You'll have to treat the snakes and the enclosures of both animals - you will never be sure that your other snake is clear of them and you'll just end up having to do it all again a week or a month later.
Forget the vet trip - the enclosures and the room will need treatment as well. Place the snakes in separate clean white or plain, pale coloured pillowcases, and remove from the room in which they live. Leave ALL substrate and cage furnishings in place in the enclosures, and leave them wide open. Depending on the size of the room, use the appropriate number of cockroach/flea bombs (from supermarket) to fumigate the room and everything in it. Leave it closed for at least 3-4 hours to allow penetration into all crevices and spaces. Once the room is ventilated, the residue these leave is not harmful to the snakes, but will prevent any remaining mites from developing into adults.
Treat the snakes separately - you can either get some Orange Medic from the chemist, dilute it 2 parts water to one part OM, and swab the snakes thoroughly with this mix, taking special care to cover the area around the head, chin, eyes and labial pits (mites congregate around these areas). Leave this on the snakes. Place into fresh clean pillowcases and check in 24 hours to see if there are any live mites moving on the fabric. This is unlikely. You can also place the snakes in plastic tubs with enough water to just cover the snakes, a place a firm-fitting lid on the tub The snakes will sit in the water with just their noses showing above the water. Add a few drops of dishwashing detergent to the water to act as a wetting agent (mites drown easily on a wet snake) and leave the snakes in there overnight (at least 12 hours) in a warm place. Remove the snakes and place in pillowcases as per the previous method. Both these techniques work very well.
Once the room is opened and ventilated, remove all substrate and cage furnishings. Replace with fresh substrate, and wash all furnishings, branches, waterbowls etc in hot soapy water, allow to dry and replace in the enclosure. If you can, use clean white butcher's paper on the floor of the enclosure for about 4 weeks, so that you can see if there are mites moving again. They will show up as pinhead-sized black dots, or a brown smear on the paper where they are squashed by the snake as it moves around.
Seems like a workup, but if you don't effectively treat the enclosures and the area around them in the room, you'll get them back after a few weeks.