Wow sounds like you and your family are in for a real adventure. If I were in your shoes, I'd go for it and I would base in Florence for the month. Lucca is a great place and I often recommend that people with small children base there instead of Florence, but they are usually only goiing for a few days or a week maximum. Your situation is quite different. The main reason I wouldn't recommend Lucca for you is that I assume you will not be having a rental car and so will be planning to make your day trips using public transport. Lucca does have good transport links, but to that part of Tuscany only and I'm not sure that there are enough day trips from there to last a month. It is not well connected to central, southern or eastern Tuscany. Florence on the other hand has the best public transport links in the region and if you use the trains and buses and perhaps join the odd organised tour, I think you will not run out of options over the month.
Secondly, it isn't true that there is nothing for kids in Florence. The Palazzo Vecchio houses the Museo dei Ragazzi. They organise activities and various tours for specific age groups. The Palazzo Strozzi puts on a series of temporary exhibitions and they also have a programme of activities to get children involved with the art, again aimed as specific age groups. The Bottega dei Ragazzi at the Ospedale degli Innocenti offers workshops where kids can have a go at making a fresco, make a model building etc. All of these programmes offer English speaking activities, you just need to get in touch beforehand and book the appropriate tour/workshop.
Then there are the other places popular with children: climbing the dome, playing with Leonardo's replica wooden machines at the Da Vinci museum near the Duomo, la Specola with its collection of stuffed and preserved animals, the Palazzo Davanzati with its maze of colourful rooms, restored to look like a Medieval merchant's house. They might like the Costume Gallery or the collection of armour and weaponry at the Stibbert. All children love shopping for tat at San Lorenzo Market. There are various parks and gardens too. Great day trips with things for kids are to San Gimignano, Pisa, Lucca, Siena, but also there is the archeological park at Fiesole, a zoo at Pistoia, an adventure park with zipwires etc. near Fiesole.
I assume you will be using the experience to help teach the children something too. What an opportunity. I would get the family ticket of the Amici degli Uffizi which gives you unlimited access to the Uffizi, Accademia, all the galleries and museums at the Pitti Palace and the Boboli Gardens, the Bargello, San Marco and more. You could do a little research the night before and then take the children to see just one painting/sculpture and get them to answer questions/find some things in the piece. Use it as an opportunity to teach them about history, art, the classics, bible stories (whether one is religious or not - they are part of our culture and history - essential general knowledge). Then just leave, go and do something fun or take a short day trip. Then do the same the next day with another piece. You can also use the month to take them on a culinary adventure trying and scoring different ice creams, pastas, pizzas etc.
Really, I think this is a fabulous opportunity for your family and with a little bit of research and forward planning you will all get a lot out of your trip. I have never had the opportunity to spend a whole month in Florence, but this is the kind of stuff I have done with my children when we have been there and hand on heart I can tell you that my kids love Florence and are always happy to be back there.
The only thing I would question is the area you are looking to stay in. Obviously in March the city will not be quite as overrun with people as it will be later in the year, but that does place you right in the heart of the city, so you will experience the noise (the worst being the street cleaning machines at night), the density of visitors and the higher prices of restaurants etc. I would look at staying the other side of the river, where it is quieter, you feel you have more space and the shops and restaurants are generally a little less pricey. You also tend to feel more like you are living with the locals than the tourists over there.