Options for Replacing 60w Magsafe Power Supply
No, you cannot just arbitrarily solder the cord from the MagSafe Power Supply on to a generic 60w power supply as the output voltage and amperage need to be matched as well.

That said, since you're having issues where it's connecting to the power supply I have to assume your starting off by wrapping the cord to tight around its ears.  You're not supposed to bend it tight at it base!  Start wrapping it by allowing it to stay sticking straight out from the power supply, as in the picture below, and then start a bend to wrap it around the ears.

My previous MacBook Pro which I had for 7 years I wrapped it as in the picture and never had and issue with it.  I still have the old MBP and power supply and it looks a good as the two year old one in the picture below.