Yes, you have to mail your daughter's amended return. You have to send the Form 1040-X (2 pages) and any other forms or schedules that are changed. If your daughter's tax return is very simple, and the only change is to say that she can be claimed as a dependent, there might not be any other forms or schedules.
Send it by certified mail with a return receipt so that you will have tracking and confirmation of delivery. It may take 6 months or more for her amended return to be processed.
If your daughter is old enough to sign her name, she has to sign the Form 1040-X amended return. If she is to young to sign it, you can sign it and write "parent for minor child" under your signature.
If you were unable to e-file your own tax return because your daughter did not indicate that she could be claimed as a dependent, you have to file your own tax return by mail. You will not be able to e-file it. You do not have to wait for her amended return to be processed before you file your own tax return.