We are going to Denali in July and are spending two days at the Mt. Mckinley Princess Wilderness Lodge and two days at Denali Princess Wilderness Lodge. Can you recommend day hikes that are about 3-5 miles long? We will not have a car but would love to avoid large packs of tourists.

Presumably you have already made shuttle reservations for the park. As the park is closed at Mile 43 because of a landslide and is under construction, this is as far as the shuttle or any buses go. You can get off the shuttle bas as much or as little as you want to. Be aware that Denali is a Wilderness park and so there aren't marked hiking trials, except around the visitors center at the park entrance. The ground around the roadway is tundra. You also definitely need bear protection. You can probably find out a good deal of this information on the Denali National Park website.

I'm guessing your transport is pre-arranged through the lodges -- correct?

Both of these lodges are very remote -- the only tourists you are likely to see are other lodge geusts. Each hotel has its own hikes and 'nature walks'

I use alltrails.com (and app) for scoping out hiking trails all around the world.

Below is a screen capture of a bunch of trails right near Denali Princess Lodge. The hotel will certainly have recommendations.
The Tiaga lake / horseshoe loop trail is highly rated (4.6mi). I

download trail maps onto my phone when I am on wifi so I can follow the trail using my phone gps. If you're going to use the app, be sure to try it out at home so you're familiar with how it works. Figuring it out for the first time in the wilderness, with no cell service isn't wise. I would also download google offline maps for the region for the same reason.

Assuming you have shuttle reservations, Savage River is a popular (by Denali standards) and excellent place to hike 3-5 miles.

Savage Alpine. It's 4 miles long. Park your car at MM13 to do the hike with a slower ascent. When done, do the Savage River trail and then take the free bus back to your car.