We spent Christmas week there last Xmas - 4 days in Mandalay and 4 in Bagan. We took our five kids who were 12, 10, 8, 4 and 2 then and travelled with my mother and her housekeeper (both live in Phnom Penh).
In order to book flights, hotels, drivers etc, we used an agency - Hanuman Travel based in Phnom Penh. Probably not much help to you but I guess there are agents based in the UK who book things there. I think in order to make a quick trip like we did with lots of kids, everything needs to be booked in advance, but maybe if you have more time you can manage without an agency?
There are wonderful puppet shows in Mandalay, boat trips on the Ayrwaddy, horse cart rides around the countryside and plenty of temples to climb (oops), all things which our kids enjoyed. We had a couple of suites at the Rupar Mandalar hotel in Mandalay both of which had two bedrooms which gave plenty of space with the kids. The hotel runs puppet and dance shows beside dinner most evenings which worked really well with the kids too - something to look at while they ate! In Bagan, it's all about the temples but our kids enjoyed the fact that everywhere we went there, Burmese kids appeared from nowhere ready to try to chat and play.
It's a fascinating place to visit but we were alone travelling with younger children - lots of backpackers and couples - and I wouldn't have liked to try to organise everything as we went along with the children as unless you know where things are, finding out takes ages, particularly in Mandalay which is very spread out ... my main tip would be to sort yourself out well beforehand! I took nappies in from Bangkok so I've no idea how easy it is to get them there. Do not let your kids eat the ice cream they sell around temple sites from large vats - made from local water - and be very, very careful where you eat with them; we had no upset stomachs but I think if you bought food from stalls, your little ones could suffer.
We didn't worry about malaria at all and I think between us we may have had two bites in Myanmar - at that time of year it's chilly at nights and I don't think the little blighters really wake up long enough to bite anyone. (Different story in Krabi where we spent New Year!) We wore fleeces and trousers, thick socks etc to dinner and breakfast, the clothes we'd travelled in from England in December; it was about 30C at the hottest point each day, so big variations in temperature.
It's a fascinating country and the people are very friendly, really keen to talk and share. There were some signs which make scary reading from the army and give a little insight about how the country is run, and we were 'escorted' throughout our time in Mandalay by a 'translator' (we thought he was a minder) but then we were a fairly unusual (and raucous) group.
Have fun!