Most of the city hotels will have steep parking fees, so it's probably cheaper to use metro and Uber. It's hard to make any recommendations without knowing more specifically where you want to be. It's been a long time since I stayed closer in than around DCA. I do have a fave hotel there, very clean, very nice staff, good breakfast included. Renaissance Arlington Capital View. I stayed recently for $99 on club level. I'm guessing you want to be closer in though, and I just haven't stayed closer in for 15 years, so I don't have a recommendation
Have you considered Airbnb as well? Having an apartment with kitchen and living space can be great when traveling with the family.