How do I connect original to new wiring when repairing an electrical fault in an AC unit?
Generally, it should never hurt anything to use thicker wire, assuming that you can make solid connections.  The wire should never be the limiting factor!  The system should already be fused so that #18AWG won't melt, so #14 will only help.

Wire nuts and tape is a tried and true method, although some would argue that soldering and heatshrink is better.

You mention "thickly insulated".  Is there any rating information (voltage, temperature, etc) stamped on the insulation?  You want to make sure the romex is at least as good.

Specifically, though, it sounds like something else is going on.  The existing wire certainly shouldn't have burnt through.  The system shouldn't have drawn more current than the wire could handle, and if it did, an upstream fuse or circuit breaker should have caught it.  I suppose it's possible that it rusted first, became less conductive, and then burned through...

If there is some sort of unprotected over-current problem, then using #14 might cause the failure point to change.  The wire became the fuse this time; who knows what wold have melted if the wire hadn't?

Anyway, just some things to think about...