We're putting in an LG all in one washer/dryer. It needs no venting to the outside, so all it requires in terms of plumbing are the water hookup and drain. I don't think that plumbing would run too much by itself, but we're doing a whole renovation so I don't know how much that bit costs. The machine is about $1000.
LG, Malber, and Edgestar seem to the ones that are most common in the US for installed ventless machines. You can search any of those manufacturers and "ventless" and "washer/dryer" and find them, I think. The Haier and others seem to be more "portable" oriented for hooking up to the sink.
That said, I think your first stop should be talking to the landlord. They might be fine with it, and even help pay for it. And your fastest, most convenient dryers like you're probably used to are regular vented ones. The
ventless ones take a long time to dry (1-2 hours), although they use less energy and are gentler on clothes.
Some concerns landlords have (the management company for the apartment we own were worried about these), washers do inherently use a fair amount of water, which the landlord often pays for. And water damage if it breaks or leaks. If they're concerned about leaks, there are membranes that go under the washer and keep leaks from reaching other floors (that's what we had to do), or alarms that will shut off the water and alert you that the machine needs repair before it starts to leak a lot.
Happy with my DH, 2 kids, dog, fish, and frogs