Since we live close by I have never don this. But, I have heard good things about Aloft Denver. I know it has a shuttle and extended parking. It's probably even cheaper than what we have to pay to park at the airport for a week
They have pretty good rates. The last time we flew (this past May) we had a later flight so instead we were going to stay at the end of our trip. It was no problem with parking either at the start or end of our trip.
We have stayed at the Sleep Inn, Clarion and the Staybridge. they were all very close and have shuttles to/from DIA. If you just want a one night, don't care what the place looks like (but it is very clean) the Clarion and Sleep Inn are cheap. If you want a super nice one bedroom with kitchen then try the Staybridge (it was the first one we tried so I paid more than I would now). They all had great shuttle service, Sleep Inn and Staybridge have complimentary breakfast (some pastries, cereal, coffee, juice).
The cost for a room is basically the same as parking at The Parking Spot (our go-to airport parking when not having to stay overnight).