There are several organic concentrates or extracts that are good for gardens and around the house and can be used indoors. Also useful for things around (literally) the house like ants, mosquitos and for the perimeter of the house for pets like fleas. Also for a double whammy one might use chrysanthemum flower varieties (check local availability or buy effective seeds and are perennial) will help. Most of these 'organic' pesticides are made from chrysanthemum extracts or synthetic products, again check the product. Neem oil is OK but not useful in bigger plants, messy, and does not last very long.
I had an indoor Meyer lemon tree with spider mites all over it, didn't notice them until most of the leaves and blossoms fell off - yes, aphids too were hiding and likely did the most damage. I probably brought them in on something else, being winter there a few ways: Put plants in a garbage bag and whisk them into the garage for treatment (above 32°) or in my case put them right in front of a sliding glass door on the threshold and quickly blast/rotate them, best of luck!
Good advice on spraying for spider mites. ‘Way back in the day when I lived in an apartment, my roomies and I washed an affected plant in the sink with a weak solution of dish detergent in tepid water, patted it dry; then sprayed paper towels liberally with the pesticide labeled for spider mites and mealy bugs, piled the towels loosely around the branches, and covered the plant in a twist-tied dry-cleaning bag for the night. This way we could control the spray from blasting outward. The fumes from the towels and the no-escape bagging did the trick.
I had to use this method recently when no-see-em bugs came in throught the screendoor and one of them gave birth inside my laptop—they were crawling out through the keys: I sprayed a paper towel with Raid, and placed the open laptop with the paper towel across the keyboard into a garbage bag, twist-tied. Did the trick overnight. Hate to think what the inside beneath the keyboard looks like now. I don’t have one of those tiny screwdrivers to open it up.