I used a different iTunes account on my phone to download apps. Now I have my own, but the apps i downloaded on the other account won't update.

iTunes keeps asking me for the password of the other account, but it doesnt exist anymore because my daughter changed her account. How do I get my apps to download on MY account?

The problem is that you do not own the apps purchased under any account other than your own.
You could potentially use the apps from another account, but you will not be able to update them on your iPhone.  The best way to try to use those apps would be to use iTunes on your computer to sync your iPhone (and hold all the apps), then, in iTunes on your computer, do Store > Authorize This Computer from iTunes' menu and enter the other account's Apple ID and password.

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