Have you tried a SD formatted to Fat32 to see if it shows up on Switch?
If it doesn't, check the pins with a light, if they are good you can open the console and re-connect the SD-reader maybe it got loose from the board connector.
If nothing works then you can try ordering a new M-SD reader from ebay, they don't cost much and are easy to replace, no soldering needed, just requires opening the console and connecting the new one.