Sounds like spider mites. They didn't come from the water, but the plant itself. They nest down in the roots or in the crotch of the plant and under the leaves. First thing you should do is quarantine your offending plant so the mites won't spread to other plants. Remove the plant from its pot. Dab the whole plant with a cotton swab dipped in isopropyl alcohol or hydrogen peroxide, especially down in the crotch and the roots. You may need a 2nd application. Clean out the pot completely with lots of soapy hot water. repot the orchid in new medium.

could also be the small white insects [name escapes me at the moment] that live in compost and don't do any damage as they feed on the compost.

Sound like harmless soil mites that just feed on organic bits in the compost.. If you don't like them remove the plant from the pot and wash off all the compost, & then repot in clean compost.

remembered the name Oribatid mite not an insect but a member of the spider group. harmless as already said. there are thousands in the compost bins come summer.