I was trying to fix my sisters flat tire to make some money by charging her fees cheaper than the bike shop.

I found the leak to be on the actual valve piece. Near the base there is a small cut looking to have generated from wear and tear. Looking at the bike wheel rim it came from the rim tape was stretched into an oval around the hole, with one side of the metal hole exposed to rub against the inner tube valve (presumably what caused the cut). 

Only parts of the rim tape were stuck to the rim so I assumed it was meant to move and completely ran a finger beneath it all the way around... On the opposite side of the rim tape next to the whole where the valve piece is threaded through is a fold where the rubber rim strip has gathered around that side of the hole as it stretched.

I tried putting some tape around the cut on the inner tubes valve stem as a quick fix, but overnight the tube deflated somewhat. 
Instead of just replacing the inner tube (I assume this would only work for a while given the issue has not been addressed) I was wondering what would be the best fix? 

Bearing in mind she will probably only pay me $10.