I have a hampton bay ceiling fan that is a few years old. It has a pull chain for the fan and a pull chain that comes through the stem for the light fixture. The light fixture pull chain has broken into the stem, I tried to remove the stem to see where the chain broke but I was unable, is it worth trying to replace the pull chain mechanism or just purchasing a new ceiling fan

Let's see, a new light is probably $95 and a new switch is $2.25 plus some labor. If you have the time and skill to do disassembly and electrical connections its well worth your effort. If you can post the model of your fan someone like Ceilingfanrepair may be able to get you a manual to help with the work. Do you also have wal switches.

Very easily done:

Ceiling fan pull chain replacement and repair - Ceiling Fans N More