Q: My sister gave me her 4S but it's still linked to her Apple ID.

Your sister needs to perform step 2 under "If you no longer have your iOS device":

What to do before selling or giving away your iPhone, iPad, or iPod touch - Apple Support

She must go to http://www.icloud.com/find then select the device and click "Erase".  Then she must click "Remove from account."

You did say "she checked iCloud."  However, there are many possible places to check within iCloud.

You did not say she checked at http://www.icloud.com/find  That is where she should now check.

So what happens when you attempt:

Settings > General > Reset > Erase All Content and Settings?

Then there is no solution for you.

Since you are not the original owner, Apple has no process for helping you gain access to the phone. This is by design and is a data security measure.

If she does not recall the password and does not wish to be of help, her phone is of no value to you.

Sorry.