OK, I just got done fighting these #$@^! aphids for over a year, so I know
EXACTLY how you feel!
I can also now tell you how to
cure a major aphid infestation.
If you're lucky enough to catch the problem early, you can sometimes get away with something simple like soapy water or pyrethrins.
But once you have a major infestation, like you've described...
my adult plants were COVERED in them. HUNDREDS OF EM. I sprayed the hell out of all my plants with soapy water, but I don't know wtf to do at this point. I even saw an aphid fly off somewhere, so I assume my house may be infested.
Then relying on these kind of approaches, you're basically screwed.
You can go on a "search-and-destroy" mission, closely examining the top and underside of
every single leaf with a jeweler's loupe and
killing every single aphid you find, and completely drenching every part of all of your plants with spray, so that you could swear on your life that every one of your plants was now
COMPLETELY aphid-free...
...and then 2-3 days later, your plants are covered with aphids again!
(speaking from direct experience here)
The reason is that soapy sprays, alcohol sprays, even pyrethrin sprays (like Bonide) are all
contact-based poisons - they only kill aphids if you spray it directly on 'em.
Remember those guys flying around? Well, they're now hiding in various nooks and cranies in your now-infested house. When you're spraying your plants, they'll wait until the coast is clear, then land back on you plants and start covering them with baby aphids!
No matter how often or how thoroughly you spray, you're fighting a hopeless battle - as soon as your back in turned, your plants are covered with aphids again! (again, speaking from personal experience here)
After fighting these guys for over a year, BeagleStorm clued me in to what proved to be the miracle cure for this problem. Get yourself some Acetamiprid. Spray all of your plants with it once. Problem solved!
Not only did it kill all the aphids on my plants within about 2-3 days after I sprayed it, but it also managed to wipe-out my infestation entirely - after about 45 days, I was able to start new plants, and didn't even have to spray the new ones at all!