So, you have a problem. You would need to have the SSN in hand, and file the amended return, before April 15, in order to upgrade her to "qualifying child" dependent and get the $2000 child tax credit (and EIC, if you qualify).
You must have the SSN before you file an "on time" return. An on time return is filed before April 15, or filed before October 15 if you have the automatic extension. But, since you have already filed, it is too late to get an extension, and for you, being "on time" means you must file the amended return with the SSN by April 15.
(What you should have done is applied for the SSN and not filed, getting the extension if necessary while you waited.)
I found a US Today story from a while ago suggesting that most SSNs are issued in 4-5 weeks, but it varies depending on where you live. If you want to make this happen, I suggest you get all your needed documents and proofs and get to the nearest social security office as fast as you can, and hope you get the number by April 15.
