First, good hotel choice. No need to rethink it; in a tremendous area at a tremendous hotel. No need to cab/metro unless you're going long distances that aren't sensible walking. But to White House, mall, Union Station, Capitol, etc. you're excellently well-off.
"Why not just split up and have Mom go to one play with one child and Dad go to another play with the other child? Kids always like having one-on-one time with a parent and with 5 days of "togetherness", an evening apart might be fun for all."
Here is a synopsis of The Two Gentlemen of Verona:
If you decide to go, sign up for Goldstar and they send you discounts for tickets. Tickets for Much Ado about Nothing (What is playing @ Shakespeare Theatre Company now) were $110, but Goldstar offered them for $29.99 and $39.99, dependent upon seat selection. The disadvantage of Goldstar is you can't pick your seats, you just pick your section. As a financial supporter of the Shakespeare Theatre Compnay, I get free tickets to all the premieres so that's typically how I go see them, so I can't comment on either of the ticket booking engines. Well, I booked two tickets through them for Much Ado because I chose to go to it twice and booked through them directly to get my preferred seats. For the Lansburg Theatre, where this performance is held, try to get J2 and J4 (also J6, J8 if all 4 of you are going) or J1 and J3 (also J5, J7 if all 4 of you are going). They are my favorites.
The nice thing is there really isn't a single bad seat in the house, but I prefer orchestra west, myself. It's kinda confusing because orchestra west is on the right, and orchestra east is on the left (From the back of the theatre) so keep that in mind if you go, for locating your seats. This website is also the ticket sales dept of the Shakespeare Theatre Company, so you can get a feel for their prices and seating here. Looks like tickets are $79 directly through STC, probably will come to Goldstar at $25 or $30 and up.
You can walk to the Shakespeare Theatre Company from your hotel (in fact, its perfectly located for going to STC, so good choosing!)
Shakespeare is good, but I don't know if you're 9 year old would appreciate it. This is a nice option if you and your husband split the kids to go to both Peter Pan and The Two Gentlemen of Verona (by the way, dress up at STC, it isn't a casual place!). However, there is another option or two worth mentioning.
1) Shear Madness @ the Kennedy Center. Tickets are $45, so a cheaper option. From Kennedy Center:
"Shear Madness is the comedy whodunit that lets the audience solve the crime! Set in present-day Georgetown, Shear Madness engages locals and visitors alike as armchair detectives to help solve the scissor-stabbing murder of a famed concert pianist who lives above the Shear Madness unisex hairstyling salon. The show combines up-to-the-minute improvisational humor and a mixture of audience sleuthing to deliver a unique performance each night. Shear Madness has been running in the Kennedy Center's Theater Lab for over 20 years, giving more than 10,000 performances and is the second longest play in the history of American Theater (its sister production of Shear Madness in Boston is #1.)"
I think both of your kids would like this over Shakespeare, especially considering their ages.
Any other questions, let me know and I'll try to help as well as I can!