How do I kill the white bugs on my hibiscus without harming my family?
You're dealing with the dreaded whitefly. Hibiscus is one of their favorite hosts. Whiteflies suck sap from plant leaves, and unless you're very astute and observant, you won't notice the damage they cause to the plant (a few leaves may appear speckled). The only way you know your plants have them is seeing them fly up when the leaves are disturbed. Whiteflies are difficult to get rid of, but it is possible!
One of the best things to do is to place yellow or blue sticky cards in your plants. These cards are coated with glue that's super sticky-to bugs and people! The color attracts the whiteflies, which fly into the cards and get stuck. You can purchase sticky cards through a well-stocked garden center or by mail. Carefully handle the sticky cards by their edges, and keep them from touching anything. If you get them on your clothing, for example, they're stuck there for life. Feel free to cut the cards in half if you think one card is too large, but do it carefully. Make one clean cut with very sharp scissors. Change the cards when they're smothered with goo-trapped bugs.
Another approach is to spray your hibiscus with horticultural oil to reduce the whitefly population. It may be impossible to completely eradicate the whiteflies indoors, but with persistence you can keep their numbers low enough to tolerate. Consider getting rid of the affected plant or heavily pruning it back before the whiteflies move to your other houseplants.