to start your machine, you want the clutch/brake pedal full depressed; the seat switch depressed by your body weight; attachment switches/levers set to off; throttle cracked 50% (if a separate throttle); choke on ( if engine is cold).

If you have short legs, be careful not to slide too far forward during start to allow the seat switch to disengage.Or, if you don't weigh that much, try adding an extra weight behind you on the seat during startup.

If it still cuts in and out as described, try replacing the seat safety switch. Of all the posts I see here on craftsmans with similar start problems that is the most common cure.

To clarify, when I turn the key, nothing happens. When it works, I can actually start the engine while standing beside it (blade of course not engaged, brake pedal out, and with the little level below the mower engagement level up and set in place). In other words, I don't have to be sitting on the seat to start it. But when this 'no start' condition happens, it does not matter whether I sit on seat or stand - turning the key results in nothing, no crank, no starter solenoid click, nothing - as if a completely dead battery (which it is not). I'm not sure if the seat swith is supposed to act this way (start without being seated), but I don't think it is the problem. Some other switch must have a bad or weak contact because when I slam down on the seat or let the mower engage level slam down, it eventually starts.

You might want to go to searspartsdirect.com and download the manual for it.
Looking at the schematic, you'll see the ONLY thing on it that will click is the starter solenoid.

You simply have to trace the circuit from where the SMALL RED wire connects to the Battery + stud on the solenoid.
Then through the fuse to the "B" terminal of the key switch.

When in the start position, "B" is connected to the "S" terminal of the key switch.
Voltage then travels through the clutch/brake switch, then the attachment switch to the SMALL terminal of the starter solenoid.

A 12V test light would be IDEAL for this circuit.

Can you tell me where the attachent clutch switch is located and how I would get to it? Thanks.

Where mowers have a deck engagement lever assembly that mounts directly onto the deck, the deck switch typically is mounted on the deck and situates into the lever assembly parts on the deck.
Where mowers have an engagement lever that is mounted on the tractor and has linkage or a cable going to the deck.........the attachment clutch switch is typically mounted on the tractor and situated near the engagement lever linkage. Looking at the IPL it appears that your switch is going to be on the tractor and not on the deck itself.

Follow the WHITE Wire

Here is another possibility. The ground wire for the starter solenoid is bolted tight to the frame, but is not making good continuity. On some of these, the ground wire is located on the rear of the mower, just inboard of the right rear tire. Reach over the tire with a wrench and loosen / retighten the bolt that connects a wire to the frame.

I have the same kind of problem- but now it won't start at all.(it does crank and every other thing has been eliminated as the problem) I even hired someone to come by, but they couldn't figure it out, either, except they said that there seems to be a short somewhere. I pulled a full wiring harness from another similar Craftsman that I use for parts, and when he switched out the ignition switch to this one, it fit the harness, Do you think it is worth trying to replace the harness with one pilfered from the other tractor? A by-pass of the kill switch failed.

There's a lot of ignition switches that have the same pin arrangement but differ electrically.

Look at the Owners Manuals for both tractors and see if the harness has the same part#..

I wouldn't replace a harness unless there was obvious damage.
I'd hone my troubleshooting skills.

Not sure why you say you have the same kind of problem?

I said that because starting was intermittent until one day it stalled and never started again. I went through the bulk of the "list" above, but nothing. I hired someone who said he thought it was an electric problem: there was a short causing something to kill the current before it starts. He tried to bypass it and installed a manual kill switch and an on/off for the gas line. We also switched out the ignition with the one from the other tractor. (It is much newer and a Briggs, not the Kohler) It started a few times and then wouldn't again. I figured that because I have been able to exchange the ignition that maybe I could exchange the whole harness. Bottom line is I don't want to buy a new mower (this one works great when it goes) but I can't afford to have someone troubleshooting at $70 an hour. I could buy a new harness, but am worried about the modifications he made to the old one.

"there was a short causing something to kill the current before it starts"

The current going where?

Without knowing the Sears 917.xxxxxxx numbers of your tractor and the one from the "replacement parts", one doesn't have a clue if you used the correct parts.
There's over 200 different schematics for Sears Tractors.