This is why the Marriott is a MUCH better choice IMHO when travelling with babies. There is a washer and drier in your unit. At DLH there is a laundry service (each article of clothing, even for babies, are about £6 each to launder).
The Cheyenne is a 15-20 min walk from the DLH, and I understand the launderettes have now been closed anyway (though most of the machines are often broken).
Now, I don't want to sound terrible here, but we have been to DLP more than 50 times, and we have gone about 20 times since becoming parents taking our kids from 10 weeks old.
The DLH is a great location with babies, but pack lots of extra clothes or buy them in the resort. Or stay at the Marriott with a car. Those are the 2 very easiest ways of doing it with babies in my experience.