We've been a few times before, usually between December and March, and the weather has always been pleasant so I was disappointed to see all the rainclouds on the forecast. I know you can never guarantee the conditions though. There was one visit to Lanzarote where we had dustclouds and (literally) a plague of locusts, so I don't know why I'm surprised.
Previous visits were pre-DS though, so the weather and related activities were less of an issue.
We're in a private villa which I think has a sky package but I was hoping to leave cbeebies at home!
We've been in winter and weather a bit mixed but we had a heated pool so rain didn't stop us. There are indoor play places and take lots of crayons and paper for DC and magazines for you. We also found rain in one part of island often meant sun in another so go for a drive and chase the sun!
The Canaries can have pretty disappointing weather in winter, it's a mistake to equate it with a summer destination and even when sunny it's often not warm in sea or pool, partly due to the wind. We have shivered in April before. Lanzarote has good roads. If you fancy hiring a car it's scenery is dramatic. Not much fun if you are 2 though. If you are in a big hotel they often have some form of indoor play area /kids club, and even the outdoor parks will be fine if you take warm clothes.
Lanzarote has quite a lot of places of interest to visit, but some of them might be a little boring or unsuitable for a two year old. However, the Guinate tropical Park and the aquarium will probably appeal the most. IMO Timanfaya National Park is a must see, but will be of more interest to the adults than a two year old.