By federal law, since you paid her she is required to give you her SSN. You can give her a W-10 to let her know that she must provide you with this information so you can file your taxes. This is required even if she is not required to file a return.
If she refuses still and you are trying to claim the dependent care credit you can mail your return to the IRS with a letter stating all of the information you have on her and your attempts to obtain her SSN.
Also, if you pay any one else to babysit, it would be a good idea to have them fill out the W-10 prior to paying them. This way, they are more likely to give you the information without a struggle.