I had a very similar machine to the link in the original post. It was a Haier HLP21E. It hooked up to the sink. (I also didn't have a dishwasher, so my days were dishes and laundry, in that order...) I also only had one baby at the time, but we did cloth diapers, and DH had a dirty job (farmer). So it got used maybe 5 days a week, and often for consecutive loads. If your laundry is currently running $130 a month, I think a machine like this is a worthwhile buy, paying for itself in 2-3 months. Not to mention the convenience of being able to do it in your own home.
If you have a sink (maybe in the bathroom) with space near it that you can dedicate to your washing machine, that would be awesome. We had a tiny apartment (399 sqft) and the machine took too much space sitting out in the middle of the kitchen. It would not fit in the bathroom. So I'd drag it out of a closet to the sink when I planned on using it.
Occasionally the machine would get a little wild during the spin cycle, and I'd have to run over and hold it down. I also had to get a $2 mesh filter to keep fiber bits out of the sink. So I could leave it, but I wouldn't leave the apartment while it was running. Also, once, the outflow tube slipped away from the sink- but I heard the water and caught it pretty quickly. After that we had a solution to ensure the tube would stay in place. Overall, I really loved the machine and was so glad to have it. I wouldn't have paid $1 per pound of laundry, but the laundromat was 2 blocks away, and the trek with a baby wouldn't have been fun, not to mention the time wasted sitting around the laundromat. I would definitely buy it again if I found myself in a similar situation. I think I used credit card points to get it, so it felt "free."
Except I wouldn't have to buy it again, anyway. 7 years later we still have it and I'm hesitant to give it up. Our current rental has a full size washer and dryer, but it's been handy a time or two when the washer wasn't working. I should really sell it. And this is so stupid, but I'm sentimental about it! Reselling when you don't need it anymore will offset part of your investment, too. I would imagine in NYC someone would love to grab that up for $100.
We'd hang dry on a small rack from ikea. Everything would dry overnight. Even the prefold cloth diapers that are 6 layers thick in the middle.