Answer: Sounds like you've imported a population of whiteflies and spidermites when you brought your ivy indoors.
Spidermites are not unusual pests of ivy. They thrive under hot, dry, dusty conditions and generally attack plants in late summer. The webbing is convincing evidence that they've taken up residence in your ivy. The best way to discourage them is to give your plant a thorough shower, rinsing both upper and lower leaf surfaces, and all the stems. You may have to repeat the shower every 3-4 weeks to get rid of the remaining generations of spidermites.
If the flying insects are white, they're probably whiteflies. If they're black, they're probably fungus gnats. Both are difficult to control, but the whiteflies are attracted to the color yellow so you can make or purchase yellow sticky traps and place them at foliage level. When the insects fly onto the yellow trap they'll get hopelessly stuck. Fungus gnats are common pests of houseplants. They live and breed on the moist soil surface. The best way to deter them is to put a barrier between them and the surface of the soil. A layer of decorative rock will keep them from living in the soil of your houseplants.
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