Hello Chelsea, and thank you for contacting us at the UVM Extension Master Gardener program with your question.

How frustrating to have spider mites!  I too grow citrus trees as houseplants and want them to be able to thrive without these types of pests. 

We appreciate you including photos and the Helpline worksheet with your note.    We agree that these look and behave like spider mites, and I have provided some links below that will confirm your theories and suggest options for treating them.

Your washing of the plant with soap and water is an excellent, non-pesticide, solution, though as you found, it is one that needs to be repeated at regular intervals to keep down - and hopefully eventually eradicate - the population.  I am sorry to hear that they have spread from your lime tree to other houseplants.   Is it possible to isolate those infected while you repeat the cleaning process?

A couple of the articles below also mention horticultural oils (which will suffocate the mites) and pesticides, and we include them here for informational purposes only.  We, as Master Gardener Volunteers do not recommend the use of pesticides, and those mentioned in the articles might not be registered for use in Vermont.

https://extension.umn.edu/yard-and-garden-insects/spider-mites

https://extension.umd.edu/resource/spider-mites-indoor-plants/

https://extension.umn.edu/news/managing-spider-mites-houseplants

I wish you success with your plants!