There are some great backpacking destinations with trailheads an hour from Boulder! Just grab a couple of maps and some guidebooks and you can get a good idea. (Sometimes easier to see things on a map than try to pull something off the Web.) Some suggestions that fit your 2-4 mile requirement are Diamond Lake in Indian Peaks Wilderness and Crater Lakes in James Peak Wilderness. Lost Lake out of Eldora/Hessie is just outside the IPWA boundary and a popular overnight destination. There are quite a few backcountry campsites in Rocky Mountain National Park within 1-2 miles of a trailhead.
You mention that everything seems to require a reservation or permit. You are correct. Welcome to the Front Range, where thousands of your neighbors want to do the same things you do. Permits are required for Indian Peaks in the summer months and Rocky Mountain Nat'l Park year 'round (bear canisters for the Park, too). Reservations are suggested if you want to go to a specific place on a specific date, but you can generally get a site somewhere if you are flexible and avoid weekends and holidays. Most would agree that spontaneity is great, but the reality of living where we do often precludes that. LNT principles should always be followed, but with millions of people in the backcountry that just isn't enough.
Get a Delorme Gazeteer for Colorado and take a look at land status. Forest Service and BLM lands are public lands that generally allow camping anywhere unless specifically prohibited. Talk to some of your neighbors and coworkers for ideas on where to go, and be open to different possibilities.
You don't say how long you have been in the area and how much backpacking experience you have, but I might suggest doing some day hikes with your daughter before you jump into backpacking. There is quite a bit more to traveling in the mountains than just throwing in a sleeping bag and cooking over a campfire: weather changes, lightning, animals, all demand respect. Perhaps you have already gotten some information, in which case good on you! Have fun.