The high peak and tissington trails are both excellent for young children as they are virtually flat and never muddy as they are of a stone/ash construction. In the high peaks case there are only 3 inclines worth mentioning in it's legnth . Not much to do on them so a picnic is highly recommended to keep the kids happy. There is also carsington water within 5 miles of that area, which is the local reservoir and has an 8 mile loop around it. Bit more challenging hill wise for the kids, but still of a good construction so no major muddy areas.
Mountains of hotels and b and b's. Try Ashbourne, wirksworth, matlock, belper, which are probably your nearest towns to the trails, or there's villages such as brassington, carsington etc. Too many to mention. Any more info required, happy to help.
Oh and we rode the high peak today with our kids of 7 and 4