Oh islands are so tough! You might go up on the NKBA (National Kitchen and Bath Association) site. They have the minimum clearances needed for a functional kitchen with an island, to help you plan.

I really blame all the media showing wonderful kitchen islands but never mentioning how much space is needed for them to work in a kitchen.

If I recall right, the amount of clearance needed from counter to island so that two people can pass each other (ie you can have more than one person on that side of the kitchen at a time) is really tough in a kitchen only 13' deep, because you also need the width of the island plus the clearance on the other side of the island...

But maybe one of the pros here can be more help, or you might go get a consultation with a professional interior or kitchen designer. Contractors don't always follow functional needs and will often give the customer what they ask for. As well as ikea or big box store kitchen planners. They just plan away, fitting in fixtures without following the design basics.

Many interior designers (designers not decorators) are trained in understanding other important aspects, like moving walls or relocating plumbing etc and may have alternatives for you.

13 feet is wide enough for a 2 foot 6 island.

If you want a continent, buy another house. Or spend a small fortune to add on a huge room to the existing house. A small island will fit the space you have. Learn to love it.

Kate,
By looking at your plan, the two walls you looking to open up are bearing walls. I'm not sure whats above and what they carry, especially the family room wall where you plan to make a 9' opening. Just make sure you consult an Architect or Engineer so the loads can be accurately calculated and they can properly size the header and something might be needed bellow if there is a basement, like a solid blocking, etc at both sides of the opening.
As the Island goes, you have room for a nice size Island and still have plenty of room left on the sides, I made an outline with dimensions of 2'6 x 7' leaving more then enough room all away around the island, in reality for your space it will be 2' x 6' most likely which gives you the option and room to add some seating.
Here is something to give you an idea with the dimensions.
By the way, very nice layout.

I did a model of the kitchen for a customer a while back, the opening from the dining room was also made bigger. You can do something like this if you like to add some design to it with a few knee walls on the side, which will give you bigger opening and at the same time will add some privacy to the family room and at the same time add some character to all that open space. The customer didn't want to have anything on top of the knee wall but if you like you can add a few columns on top of them to bring the effect out even more.
Here is something to consider instead of having a plain opening.