My MacBook Pro (2010) doesn't charge
I know there's a question mark over your charger, so it's understandable you may not have tried this, but it's still worth at least trying an SMC reset. 

To do this on your MBP, follow these steps:


Shut down your computer
Keep the MagSafe adapter (or 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.


If nothing happens, you could use the MagSafe lead and connect it to one of the newer MagSafe 2 adapters you have (keeping the lead but changing the power source) and then trying the SMC reset again.

Let us know how you go.

[EDIT]

Ok, your MBP 2010 has a 60W Power Adapter and so does the MBP 2013. 

If you can borrow that (the MBP 2013 adapter), then connect your MagSafe lead to that and connect it to your MBP 2010 for a while and leave it. Take note of whether the power/charge light comes on or not. 

Regardless, leave it connected for at least a half hour and then try starting your computer. If it boots okay, then check the battery status (in the menu bar) to see how much charge it has and whether there's any sort of error message. If there is, take a note of the message. Then shut down your computer and leave it connected to power for a while (if possible, long enough for what a usual charge would take).

Finally, before having to return the adapter, redo my steps for resetting the SMC.

Once you've done all this, report back on how this all went (any power light, any charge, any error message, etc).