Oly - I would question your (admittedly not very specific) dedication to "Burgundy". I don't suppose your kids are seriously into wine? Burgundy is fine, and on a leisurely trip around France for more experienced travelers would probably satisfy (especially foodies). But IMHO, for a first trip to France, if you're looking for someplace in France to spend a few days (other than Paris), I think the Loire Valley and/or Normandy pack a lot more "wow" for most folks.
On my first overseas trip with my then-girlfriend/now-wife, I dragged her away from Paris to Normandy, Mont Saint Michel, to several of the more jaw-dropping Loire Valley Chateaux, Avignon, and Carcassonne as the cherry on top. She was blown away (probably why she married me) - even though we spent a bit of time driving around France in our rental car. It was an awesome trip. If I were you, I'd skip Burgundy and try to slip in some of the places I listed above - they are very special and your kids won't ever forget them (neither will you).
Although many people would disagree with this, I try to maximize the benefit of a trip to Europe (or anywhere) by doing the things there that I can't do at home. Hot air ballooning? Yeah, I'm sure it's nice in Tuscany. But I can go hot air ballooning in Snohomish for a fraction of the cost, any nice day, any time (and it'll be pretty nice). But seeing the sun through the stained glass windows of Chartes Cathedral? - can't do that at home for any price. You can ride your bikes in Yelm any time you want, but seeing the tide come in at Mont Saint Michel is (for most of us) a very magical, once-in-a-lifetime highlight of any trip. I have nothing against doing fun activities while away, but I try to remember which things I can only do in Europe - there are more of those than time available. For me (and I suspect for most), the most limited commodity we have is time-in-Europe. Be sure you take advantage of that time as much as you can.
No car rental? OK, just know that by making this declaration, you really limit your options. Sure, there are lots of places where having a rental car is just an expensive headache, and you really don't want one there. But in other places a car opens up wonderful opportunities. What a shame it would be to miss great places just because "cars are bad."
You are 100% settled on round-trip to London? OK, but...you are burning a couple days by sticking to that. Flying home from Paris effectively buys you another day or two. What would you be willing to spend to get two more days in Europe? Depending on hotel costs (London is super-expensive), and LHR has the highest departure taxes (built into your ticket price), and the cost of getting BACK to London...you might actually save money by re-thinking (even changing) your flights. Worth a look.
Hope some of that helps. Have fun.