Electrical – Anybody know how to change the bathroom light bulb case/cover

it's usually pretty easy to change the fittings:

• make sure electricity is off (turn off the breaker that powers the light)
• double check the electricity is off
• loosen the screws that hold it up on the ceiling and keep the wires attached. Remember which wire is connected to the ground (bare/green or yellow green striped wire), live (red, orange or blue wire) and neutral (black wire) connections. If your house is newer then the colors may be different: green/yellow stripe for ground, brown/black or gray for live and blue for neutral
• connect the wires as indicated in the manual for the new fitting
• attach the new fitting to the ceiling
• put in new bulb
• turn electricity back on, if the breaker doesn't want to stay on you did something wrong and should inspect what you did wrong before trying to "fix" the breaker.
• toggle the switch and bask in your accomplishment.

I found the answer! @diceless set me on the right track in looking at where the foot of the bulb is connecting to the fixture.

Since it was an older fixture (installed in the 80's, I think), the hot contact (Number 1 in this diagram ) was bent down, and it was connecting with the smaller bulb but not the bigger LED bulb.

With the circuit turned off at the breaker I reached in with a wooden tongue depressor and bent it upwards slightly. Then, I made sure to screw the LED bulb in tightly. Flicking the switch it was bright as day!