One recessed ceiling light has been broken for many years. So I decided to fix it. From a book "Popular Mechanics Home How-to":

It is too old. I checked mine:

To figure out what happened, either the wire or the socket, I have to move  every thing out from the ceiling fixture.

1. There are two spring to hold the plastice cover on the fixture. To take the plastic cover off, remove one end of the spring from the hole by lifting up the upper end of the spring. Remove the other spring's upper end too.

2. Remove the butterfly screw:

3. To move A from B, push two pieces of C towards each other:

You will get an inside view:

4. Move 3 screws off so that the whole fixture is off from the ceiling:

The fixture is off from the ceiling.

6. There is a thermol fuse:

Take the thermol fuse enclosure off from the back of the fixture with these three pieces:

7. Measure the conductivity of the thermal fuse with these two attachments to the probes of the multimeter by pinching the two wires connecting the thermal fuse:

8. The theomal fuse is not conducting, so it is broken. I used a 128°C thermal fuse to connect the two wires (I have ordered 10 pieces of  65°C 10A thermal fuse from Aliexpress, and will replace the 128°C thermal fuse)：

The connectors used are:

9.  Put everything back in the reversed order， the recessed ceiling light is now working：

Note: I have ordered 10 pieces of  65°C 10A thermal fuse from Aliexpress, and will replace the 128°C thermal fuse.