This might not be your problem, but I see an opportunity to share some information.
I can't swear for this, but I have been told that when you give someone a "jump start", turn your engine off or you run the risk of burning the diodes in your alternator.
I am not big on electrics/electronics so I am not sure of this, but supposedly the large power draw of the starter can cause what they call a soft failure of the diodes, where they don't fail completely, but over time the alternator life could be shortened.
It might fail tomorrow, or it could fail nine months later.
This is the recommended procedure:
Connect the cables, run your engine for a while to get some charge into the other battery.
Turn off your engine.
With the cables still connected, try starting the other car.