Two is a great age for WDW. My son had such a great time there in late April/early May -- he still sings It's A Small World almost every day.
I wouldn't wait to go to the Magic Kingdom, though-- we spent part of every day there (we had ultimate Park Hopper tickets), usually the morning. If you can make a PS for a breakfast in the park before 9, you'll beat the crowds through security and the turnstiles, and you can be waiting to get on the first ride of the day as it opens. We chose a different "first" each day-- one day it was Dumbo, another it was the Tomorrowland Speedway etc. Obviously you will spend most of your time in Fantasyland-- the Carousel, Small World (if your daughter is like our son you will ride this several times in a row, every day! but he loved it so much and the vacation was for HIM not as much for us so we did what he wanted to do-- the secret to success, I think!). Dumbo, Teacups, Winnie the Pooh, Peter Pan-- all were good. We did not do Snow White-- too scary. Matthew loved Jungle Cruise. We skipped Pirates after riding it ourselves to check-- much scarier for little ones than I'd remembered. Definitely skip the Haunted Mansion. Country Bear Jamboree was a favorite. he did not like the Enchanted Tiki Birds-- scary Tiki God and thunderstorm ruined it for him, plus it was in the dark.
Take bubbles with you-- a great distraction in line. Also take your own little cooler with drinks (if you rent a fridge in your room, you can freeze juice boxes the night before and they thaw as the day goes on, so you can avoid a long line when she gets thirsty) and snacks. Take stickers or a mini Magna-Doodle to keep her busy too. Some lines, whether for rides or food, are probably inevitable.
Don't miss the Animal Kingdom if she likes animals-- our son loved Kilimanjaro Safaris and the walking trails. The character breakfast there at Donald's Breakfastosaurus had great food and was a lot of fun too. We went to many character breakfasts and dinners. In fact, we started our trip with a firts dinner at Chef Mickey's in the Contemporary-- he was a little shy with the characters when they came up to him, but he loved watching everyone and joining in with the napkin waving and dancing.
Some restaurants are great for kids-- try Epcot in the early evening and then have dinner there. The Teppanyaki Dining Room in Japan is fun for kids because they can watch the food being prepared in front of them. We found that the fireworks were too intense close up-- but when we watched them from the beach at the Polynesian, our son really enjoyed them--not as loud from there. Other restaurants: Boma in the Animal Kingdom Lodge, 1900 Park Faire at the Grand Floridian, Kona Cafe in the Polynesian, Concourse Steakhouse and Chef Mickey's in the Contemporary (don't go to Chef M's for the food-- it's strictly for the ambiance there!). Crystal Palace for breakfast.
I know I am rambling-- sorry! Trying to cram a lot in! Definitely take your own stroller! You need one outside the parks too-- at your hotel, etc. We spent a lot of time at our hotel pool (at the Polynesian) which our son loved to do. We found that there wasn't much for him at MGM Studios, but don't miss Playhouse Disney if your daughter is a fan of Bear in the Big Blue House. There is also the Honey I Shrunk playground there, which is fun.
You already know that the NAP is crucial to your well-being. That's the biggest mistake people make. But if you go at your daughter's pace, you'll have a wonderful time. Some of the best times are not on any attractions-- my son loved watching people come down Splash Mountain and getting wet-- he laughed at all the cars coming down for about half an hour.
HAve a great time!
Don't miss Toontown in the MK, or the railroad ride around the park-- a ncie way to cool off. Parades are good, but get a spot in the shade!