Laundry shoot would be the answer in my house - sounds like it would be in yours too :)
I have mobility issues so in our house, everyone is expected to bring their laundry to my laundry room. We had to use hampers so that kids would actually put their laundry somewhere, but when full or when mom is doing a wash - it is one of my kids' duties to run around and get it for mom.
When I do sheets or towels, each kid (husband helps the little ones), brings me their sheets.
We have a drop down our stairs (just open stairwell) that lands near the laundry room. So if I'm home and can do laundry, I chuck it all down the stairs.
Is your older daughter able to take her laundry to the laundry room? I have baskets on floor there - kids will biff their stuff into them. It's habit now, so it's not a big deal for anyone, and it's tied into their allowance.
Good luck :)
I'd love a laundry chute though ... don't they sound great? Wonder why they stopped putting them in.