Can you reach in with a rake or 'scuffle hoe" or regular hoe? Scraping them off at ground level frequently might discourage them enough for chemicals to have some effect. If nothing else, keeping them small keeps them from growing through your deck's boards
After pulling out as much top-growth as possible, some weeds will be further weakened by denying them any light or water. Cover with thick dark plastic and weight it down so no light or water can reach new sprouts.
If you can reach in with shovels, you could excavate 6" to 12" of soil, and screen the roots out of it. Don't replace the soil under the deck! Instead back-fill with rocks or gravel. Or concrete.
Assume that the excavated soil still has rootlets and millions of weed seeds. Don't spread it over other beds, keep it in one place, perhaps as a small raised bed that will be easy to pull weeds out of. If the weeds liked shade, locate it in full sun.
Amend the soil with something that makes the soil loose enough that it will be easy to pull new weeds that sprout. Never let the weeds get big enough to go to seed.
Plant aggressive things densely enough to choke out even weeds (vegetables or cover crops or weedy fast-growing flowers). Mulch heavily.
Give the "weed seed bank" enough years that all weed seeds have sprouted, decayed or been eaten by worms before you consider spreading that soil back into any bed that you care about.
If the house and deck are really, really REALLY well-insured, you could try a blow torch under the deck, but this does NOT sound like a situation where fire is the answer.
