Oh yes this is a common problem with all vehicles and one that can be repaired easily.
First off some background info to get you in the right mode. The speed control chooses the speed of the motor by changing the resistance in series with the DC fan motor. These resistors are all attached to one board with shovel plug attachments and the whole assembly is stuck in the air vent to keel them cool. The motor drives the air to cook its own resistors. The wiring to this unit has the connections housed in a plastic plug matching the connections on the board. Now where is this resistor board in your car. Well it is directly on the other side of your glove compartment shelf. You can pull the drawer down by carefully dislodging the stop clips found at the upper edge of the drawer on either side. Squeeze the top sides in a bit to allow the compartment to swing out and down to reveal the wiring behind. Now the resistor board part is fastened to the air vent with two Philips screws. Unplug the wire connector carefully using a small flat blade screw driver to move out the latch type arms holding the plug into the resistor board. Then remove the two screws and pull out the resistors.
In most cases you can see the burnt resistors but also this connector can be burnt black and melted, while the resistors are intact
To solve your problem you can purchase either the resistors , the plug harness or both. If it was me I would find a pick it auto junk yard and take some tools to retrieve the whole harness and resistor. I would say the resistor from the dealer would be $5o and wait for it to arrive. The wiring repair plug and harness probably another $50. Pick your used part probably $10.
Oh yes, take the old part with you so you can match the parts. You will find this fits other year Honda cars so have a look before passing a possible vehicle. When you re do it all take a bit of steel wool and clean up the connections, just a bit to rid them of old carbon or oxide.