I have a large area covered in mile-a-minute plants. They have been spreading fast and killing out the other plants. How can I kill them permanently without hurting other plants?
This site indicates you have a major problem that will require large amounts of effort no matter what method you choose. You can hand pull while you are wearing gloves and try to get the root. This could take years if you have open soil and moist conditions.
If this was my weed I would write off the plants around it. Cut everything down, remove and lay down a thick geotextile or landscape fabric and cover with 2 to 3 inches of mulch. Leave on for several years.
I remembered something I read here from a wise gardener whose name escapes me. You can kill some weed seeds with solarization. By removing all growth and covering with plastic during the hottest part of the year you could get rid of some of the weed seeds. Further work is likely as you will not get all the seeds in one go.
The other answer talks about how to kill the existing weeds and seeds (which is what you asked), but you also need to consider preventing future problems.
Where did the weeds come from?
If your population came from an infestation in a nearby roadside ditch or riverbank, it will continue to throw seeds into your area. Controlling your immediate problem will certainly help by reducing the seed bank. But also consider prevention of future problems by looking around to see what could be contributing to your problem.
I mentioned this in another answer specifically about ragweed, but most of which really applies to weed control in general:
Seeds can be dispersed by wind spreading them along the soil surface. This means that if you have uncontrolled ragweed in another part of your yard, have neighbors with uncontrolled ragweed, nearby fields, or if your town doesn't keep the roadside ditches mowed, then you'll end up with more seed in your garden year after year.
With mile-a-minute, birds are another vector that you can't control. You'll have to be constantly vigilant and attack any future weeds that pop up before they have to a chance to spread out of control.
Mile-a-minute mocks solarization. It will bust through a black multi-mil sheet in midsummer. Hand pull or chemically kill. I do both. Hand pull first, kill what grows back.