How much does the compact crank that they're willing to install for you cost and what else, if anything, might they have to replace to make the compact work? Depending on the answers to that, $220 might be a pretty good deal--assuming the compact will give you the gears you need. I certainly wouldn't expect them to do it for free.

I know you're frustrated, but I have to say that I disagree somewhat with laying the blame for this situation solely at the feet of your shop. Ideally, they should have talked to you about gearing, but as certain responses to this thread have indicated, there are women out there who ride with a standard double just fine so your shop may have had no reason to question you on it. I'm sorry to sound unsupportive or unkind, but given that you're not a new rider, I think you should take at least some responsibility for having bought a bike that doesn't meet your needs and, as such, bear at least some of the cost for fixing the problem.

Now maybe your relationship with this shop is already beyond repair, but I would suggest taking a few steps back from the situation and a few deep breaths before you do anything more. First and foremost, really figure out what gearing you need. If you still have your old bike, take it for ride on the hills you frequent and see what gears you're using. Make the ride long enough that you climb some of those hills while fatigued. From there, get a firm understanding of what choices you have in order to meet your needs and how much it will cost to make the required changes. Negotiate from there and do what you can, i.e., by selling whatever parts you no longer need, to lower the cost.

I hope the situation resolves itself to your satisfaction soon.