My ceiling fan works on high but the mid speed is now the same as the low speed. I changed the switch but now the mid speed doesn't work at all. Is there another part I need to change? Thanks

The likely problem is that one of the capacitors changed value because that's how the speeds are changed. The switch selects 1st and then the 2nd and then the parallel combination.

Replacing the switch should have given you the same symptoms. Besides I can't tell from here what actually happens in all 4 positions. Like buzz, stop, seems like high.

The dual capacitor is the likely problem and it's probably rectangular. CeilingFanParts.com Home is a source.

Is it the part that has 2 wires running to it and soldered on the ends. When I replace it, do I have to solder the connections?

It should have 3 leads, like these: CeilingFanParts.com - Switchco Products - Capacitors

And have markings like 6 uf 250 V/ 3 uf 250 V.

The speed switch should go directly to 2 of the capacitor leads.

The soldered thing could be a thermal overload.

Yeah, put the original switch back.

What make/model is the fan?

Ceiling fan capactors - troubleshooting and resources - Ceiling Fans N More