MacBook Pro now asking to login with password, but can't do this via built-in keyboard
After reading your question a few times I've made an assumption (dangerous, I know). My assumption is that the only password you're being prompted for is your daughter's User Account password and NOT a firmware password. The reason I assume this is because the only reason (it seems) that you think it's prompting for a firmware password is because of Adriano's answer to another question.

So, if my assumption is correct, then my first thought was the same as that offered by n1000 in the comments - Have you tried an external keyboard?

However, if you haven't got an external keyboard handy, and since you're booting into Recovery Mode anyway, you can actually change your daughter's password from Recovery Mode and then log in using that and change the settings so a password is no longer required. 

Why would this help? Well, since your Caps Lock key is stuck, creating a password with how your keyboard is right now will mean that you will be able to log in using that keyboard.

Resetting password from Recovery Mode

To reset your daughter's password password from Recovery Mode, follow these steps:


Power off your Mac
Boot into Recovery Mode by pressing the power button while holding down CommandR
Select Disk Utility and press Continue.
Choose Utilities > Terminal.
Enter resetpassword (all one word, lowercase letters)
Press Return
Select the drive with your daughter's account (typically the main hard drive)
Choose your daughter's account using Select the User Account.
Enter a new password 
Re-enter the same password
Enter a password hint
Click Save 
Ignore the warning about the password changing
Click OK.
Shutdown the Mac using Apple > Shut Down
Restart the Mac normally and use the new password to log into your daughter's account
Once logged in, go to Apple > System Preferences
Click on Users &amp; Groups
Click on Login Options
From the Automatic login pop-up menu, choose your daughter's account


This will ensure that whenever your daughter's Mac is booted it will automatically boot into her account (which it sounds like you want to do).

If I've misunderstood your question, or you have any questions/issues, let me know by posting a comment.

[Edit]

Now that we've got you logged in, let's try sorting out your keyboard issue. Since you can't reset the PRAM as per usual, it's actually worth resetting the SMC instead. Sometimes this will resolve the issue preventing a PRAM reset.

For your MacBook Pro, here’s what you do:


Shut down your computer 
Keep the MagSafe adapter (power cable) plugged in. 
Press at the same time shiftoptioncontrol 
(on the left side of the keyboard) and the power button 
Let go 
Turn your computer back on with the power button. 


Once you've done this, log back in and see if there's been any change to your keyboard and report back.