Buying a gym membership doesn't do anything for you. You actually have to go there and use the facilities. Are you really going to do that or is this a New Year's guilt move to assuage bad feelings about your health.
The other point I'd make is that weight gain is mostly about what you put in your mouth. Working out has other benefits, but if you are not disciplined about what you eat it won't result in long term weight loss. Just look at calories burned at various gym activities and what they relate to in your regular diet. You can eat back calories burnt at the gym faster than you can work them off.
Yes we both used our memberships before and had fantastic fitness levels. On average we went 5 days a week burning off 500 calories each workout, 50 weeks a year.
So for the two of us our work outs added up to 250k calories a year, so if 3,500 calories equals a pound then we were burning off 71 pounds of fat a year. So I don't think our diet would conflict with the weight loss as long as we do what we were doing before and not taking it easy in the pool with our son every day.
There are a couple of things that might be going on here, in my experience.
Pre-kid, for sure, my husband and I went to the gym more. We went together. I'd say we went to the gym 2-3 days a week, together. I went on the other work days by myself. He biked to work 2-3 days a week. I biked once or twice.
Kids change that a lot. For sure, the 110 pounds gained - some of that is what you eat, but a lot of that is the kid - pregnancy, for your wife, plus stress, plus lack of sleep. Even if you kept everything the same - activity levels and food intake, but cut your sleep? You'd gain weight.
So...
Working out at home is great, if you do it. It can be super duper hard with kids. I really can't do it if my toddler is awake. He will not let me. This morning, he woke up 5 minutes after I did. Also, it's super easy for the "lazy" parent/ spouse to make it hard on the other spouse. If my husband doesn't get up on his workout day, then I can't just get up, turn on the lights, grab my clothes, and work out.
I am personally a social animal, and an anti-social animal. I have a gym membership (two actually). We joined the YMCA when my older son was 5 for the pool. I'd gotten injured running and started swimming. In a good month, I swim 6-8x a month. In a bad month, 2x a month. In a good month, we take our kids 4x. In a bad month, 0. The pool is really really great for kids, as is the YMCA in general. I highly recommend it. But I don't know if I get my $102 worth. (In December and January, we went 2 times each month. However there are months we use it a lot more.)
I personally found the second child made it harder. When we had just one, we'd go to the pool mid-week one night and one weekend day. Especially helpful while pregnant. After the second one, there was a while where we just didn't go (water too cold for the baby). Then of course, we are one-to-one parent-to-kid, so it's fun for the kids but not exactly exercise for us anymore. And now our older son is in baseball, so there are time conflicts.
I would say that if you were regular gym goers before, and you find it hard to stay motivated to work out at home - then I would join a gym. Sometimes, not being able to work out at home is because you are lazy. And sometimes, you aren't "broken", you are just someone who requires external motivation. I wish I could be motivated to work out at home 100% of the time, but I'm not. I have a bunch of workout DVDs and equipment. I love using them. I like to go for walks. I like to bike.
But I REALLY need those 45-minute stretches in the pool, to lose myself in my thoughts. Without distractions "I'm hungry", "I want the TV", "stop working out" (or just flat out having a kid sit on you). That's where the "anti-social" animal part comes in.