The reason it quits working on all except high, is High is direct blower to switch (possibly in one terminal and out another on the resistor but not using the resistors) while all other speeds utilize one, two or all three of the resistor coils. The reason it doesn't work is that round silver thing you see in the pics just behind the resistor coils. That is a thermal fuse. If it gets too hot, it blows, and you get to replace the resistor.
I had this same setup on my '99 Plymouth Breeze. First time I went to the dealer for a new one, he reached behind the curtain on one of the aisles and without looking grabbed a box, which was a resistor. That told me right then, they sold alot of them. When the second one blew, I started thinking long and hard about this.
Q) When is the resistor hottest? A) when it is on the lowest speed.
Q) What cools the resistors? A) The air from the blower passing over them.
Q) When is the LEAST air passing over the resistors for cooling? A) when the blower is on the lowest speed.
Q) What can make the resistor even hotter than normal? A) When you leave the switch on low when you switch off the engine, and go to restart the engine, the voltage drops, thus the amperage rises (amps and volts on a fixed load have an inverse relationship). The resistor gets even hotter than normal operation, and the blower is barely running, just starting up, and at a slower speed with the lower voltage of the start.
Easy cure is to ALWAYS turn the blower off when you shut the vehicle off.
Mine both blew in warm weather, as the blower is blowing hot air over the resistors at the instant of start up, while in the winter, the air is quite a bit colder generally.
After I blew the second one, I took a piece of #14 solid copper house wire, stripped it bare, and tightly wrapped it around the legs of the now defunct fuse, and used my Wen 100 watt soldering gun to flow solder over the wraps. After that, I made it a point to always turn the blower off.
Later, just before I sold the car, I was concerned about the resistor getting too hot. So I removed it, traced the connections and determined which of the coils was the one used only in the low speed position, and clipped it off. This left me with med-low, med-high, and high, just not that super slow, low speed. Took a black marker and marked out the white dot that represented the low speed position and that was it.
Charles