Nothing happens when I enter new nozzle temperature

Partway through a multi-colored print, it said it failed to pull out filament, likely due to a clog or broken filament in the extruder. All 4 colors are retracted and remove easily, so I figured there must be a clog left behind in the extruder. But when I try to bring the nozzle temperature up to 260 so I can unclog it, it won’t move from around 40°. Why?

At a guess, the reason the nozzle clogged was because it wasn’t staying hot anymore. And that’s why it won’t heat up now. Your nozzle heater may have croaked.

Have you tried to power cycle?

It seems like a bug caused by moving/touching the extruder with the printer on.
Reboot. If the issue persists, do a self-check-up to get the error code or check if the cables within the extruder are connected.