Cuppla things you might do w/o spending a bunch... Take new battery back out - take to a shop that can perform a LOAD test on it (new L&G eqpt batteries are notorious for being bad as new). If battery fails test, take it back for exchange and require replacement to be tested in same way before putting it in tractor. Be careful that + and - terminals are correctly connected.
While battery is still out, take a bit of sandpaper and clean all electrical connection points to bright metal, inlcuding where Batt ground wire connects to chassis frame and large wires hook to solenoid/starter.

Remember - when trying to start, clutch/brake pedal must be fully depressed and PTO switch/lever set to Off.

If no "click" from solenoid/starter then, closely chk all connections at rear of ignition switch (Caution: take before/after pixs or mark wires first to avoid mixing up connections) for good connections/no corrosion.

Post back with findings - we may be able to help further.

The battery may be fully charged, but a bad battery won't have any capacity. 8 penlight batteries will produce 12V, but certainly won't start the engine.

Did you use the same solenoid and connect the small wires properly?

You can jump the 2 large terminals on the solenoid to see if that will crank the engine. That would tell you if the battery & starter motor is good.

IF this engine has a fuel solenoid on the bottom of the carb, it should CLICK when you turn the key ON. IF it doesn't, check for a blown fuse.

The wires on the back of the ignition switch is a coupling type plastic box that is plugged in. It doesn't seem to fit tight on the ignitions switch but I was unable to get it to go closer. There isn't any corrosion that I can see.

Also couldn't find a fuse anywhere near the fuel tank. My solenoid is under the seat. You have to disconnect the battery because the wires are to short to pull the battery out to get to the solenoid. There isn't any room to jump the solenoid because you can't get the jumper cables on it and the battery wires are so short that I would have to lay the battery on its side to hook it up while it is out of the mower.

I appreciate everyone's time and help.

Can I take the solenoid out of this compartment and try and jump it safely?

***"Can I take the solenoid out of this compartment and try and jump it safely?"***
I don't advise trying that as the solenoid mounting bracket probably serves to ground the solenoid to the chassis and without a ground the solenoid will not work.
If you have some booster start jumper cables you can safely set the battery out of the tractor and connect the red jumper cable to the tractor positive cable and then to the battery positive post. You might want to wrap up where the jumper cable connects to the tractor battery cable with some type of insulating media (plastic trash bag maybe) to keep from accidental shorting on the chassis ground. Next.......connect the black jumper cable to the battery and lastly....connect the last black jumper cable end to the negative battery cable end or to a clean bare metal chassis bolt or other ground.
Now you can jumper the solenoid by touching a small jumper from the battery cable side of the solenoid to the small wire post on the solenoid.
A safer way to test the solenoid is to use a volt meter or 12 volt test light on the small post/wire of the solenoid while someone holds the brake pedal down and turns the key to start position. If the small post/wire shows voltage at that time there is a good chance the solenoid could be bad.
If the small post/wire does not show voltage at that time then the brake safety switch or the key switch are suspect.
IF....the small post/wire shows voltage and the solenoid has clicked (operated)..........but the starter does nothing.......then you can jumper directly to the starter cable (the OTHER big solenoid post) with the red jumper cable instead of the battery cable side of solenoid.
Just do the same procedure as first except connect the red cable to the starter side of solenoid or to the starter post itself. Touch the last black jumper end to a chassis ground to test starter. Caution.....it might spark some when you make that last contact. If the starter operates then the starter is OK.

Does it have a carb solenoid? IF SO, does it click when you turn the key ON?

I don't know what a carb solenoid is? I am a fairly capible women and can usually fix just about anything that breaks. It has been the only way I can make a go of things. This lawn mower is winning but I'm not giving up. Can you describe it to me and where to look. The solenoid I replaced has two post and a prong in front that a wire plugged onto. I took the wires off one post at a time so I wouldn't inadvertantly put something back wrong.

I looked at the owners manual for your tractor - appears it does not have a engine carb shutoff solenoid mounted.
M/W, take your time - closely read Mownie's suggestions on how to check the starting circuit and solenoid - let us know results.

I think what Bill was asking is to sort of give us an idea of whether ANY electricity is getting out of the key switch to operate the other things on the tractor. Lots of modern (late model) tractors have a little fuel solenoid on the bottom of the carburetor. The fuel solenoid itself has nothing to do with the starter actually cranking.................but knowing it had a fuel solenoid that clicks properly each time the key switch is turned to run (and back to off) would tell us whether those circuits are being fed from the key switch at the proper positions of the key switch.
So far I am getting a feel that there was not a problem with the starter solenoid at the outset of this adventure.
Doing a thorough test of the brake switch and the deck switch is going to require some more info.....and patience....from you, along with knowing how to use at least a 12 volt test like or a DC Volt meter.
Game on?
Does this tractor have an electric deck clutch (flip switch or pull switch on dash) or do you engage the deck by operating a manual lever?

You are right it was running fine when I realized that the deck was not disengaging when I pulled the lever all the way up. After I shut it down that time it didn't want to start. So I replace the pressure sensor( is what I call it because when the lever is all the way up it puts pressure and pushes down a button that is suppose to disengage the deck)then when I tried to start the mower it tried once and then all it would do is make a clicking sound. I then got online and most of what I read said the clicking was a bad solenoid so I replaced that. I also tried to jump start the mower with my car once that didn't do anything. Now turning the key does nothing noticeable, no clicking, the engine hasn't even attempted to start. I have a manual clutch that I have to push in and have to pull up a small lever for the choke. I do own a volt meter but just moved and will have to find it. I think it is in the garage in the tool box. Should be able to look for it Monday, today was my son's 18th bd and was busy with that. I also have plenty of patience and appreciate yours in helping work this out.

It is odd as I have this same tractor and been having similar issue. Mine will not start using key. I changed solenoid as well. Still does nothing. I changed the battery. No change. Im pretty frustrated as well but not giving up either :-)

Hi Keith did you get your mower to run?

i have the same issue mower was running now it won't start, battery is fine can jump start straight to the starter and it will start no problems. When you turn the key you get a clicking noise but the engine does not turn over.

Any help is greatly appreciated

Yes I did. It was the safety switch on base next to gas tank just inside the frame. Replaced it and have not had a problem since!

Thanks Keith mine was a solenoid under the seat next to the battery had a broken wire repaired working great now

cheers glen