I'm probably qualified to talk about this as I currently am the one working from home (mostly) with our kids who are 3 and 5 (oldest will start Kindergarten this year). Here's what I've found:

The amount of work I can get and do is not limited by what people are willing to throw at me. It's limited by how much time I can focus on work. Kids are wonderful, but they are a big distraction. And they have no concept of the mental ladder we have to climb every time we "get in the groove" programming. Some tasks like handling a release or doing maintenance on something I'm intimately familiar with I could do easily with all the distractions. Other tasks that require a higher level of concentration simply have to wait until they're in bed, or until I find a way to get away from the kids for a few hours.

I'm not concerned about being stuck in daddy mode. First, I enjoy it. But second, I realize that while in this mode (a couple more years before the youngest is in school), I just have to focus on work-related things that are compatible with how the demands of being the primary care-giver constrain me. So I focus on things that are compatible, and trust that when the time comes, I'll make the transition back without too much trouble. Keeping up with the state of affairs does help, I believe.

Don't lose your contacts. And do recognize that they grow stale. This means you have to keep cultivating new ones, and maintaining existing ones. Part of this comes from continuing to work, even if it's just enough to keep your chops. Also look at open source stuff, which tends to have less firm deadlines. Part of it comes from participating at the community level. ...that's probably a big part. Get involved in local organizations.

Don't stop learning during your time off. Take it as an opportunity to learn a new language, or several... new databases, new ways of looking at problems. If you can use your time away from programming to become a more well-rounded programmer, your time away won't be lost, and you won't feel burned out.

Update: One strategy I use is this. It's easy to get into the habit of wanting to spend all ones time with work, or all with family stuff. It's hard to draw lines. So I make it a point to say, "If I can get XXX hours to work on YYY, then I can devote the rest of my time today to ZZZ." Then I find a way to make sure that I get XXX hours to devote to YYY. If I don't do this, I find myself getting stressed out, frustrated, and grumpy. Much better to know, "This is daddy time.", and "This is work time." than to try to muscle through each at the same time. I really find that being able to get away for a few hours to focus on "work" helps me to be happier in my non-work time as well. But I also have to be willing to make a change of plans as things come up.