How do I start my daughter's return now that I have finished mine? Certainly you don't expect me to open a new account for this do you?

Yes, you need separate accounts.   For 2 people to do 2 different returns, you must have 2 different accounts, separate User IDs, and separate fees (if applicable).

If you try to prepare a second return in the same account, then it overwrites the first return making it unrecoverable.

You can use the same email address in the Account Settings (Profile), if desired, for notification/administrative purposes, but each User ID must be unique.  A User ID does not have to be an email address.

For future reference, the desktop products (CD/download) can handle multiple returns.

Please see this FAQ on how to start a new (second) Online return: