13:13 is crazy. $15-$17/hr in your area is a very good rate, especially since you have no housekeeping or additional duties expected.
No, not crazy - I just know what "you get what you pay for" means. A college educated nanny, with experience, will cost between $20 and $25 an hour.
Not true. Very few college educated nannies for a single child in DC is earning $20+/hr, especially if there is no housekeeping in the job description. The average market rate is $15 to $17/hr. OP can research this in a number of ways, including polling her friends and neighbors with nannies and posting on neighborhood list serves. The nannies here routinely try to inflate rates, which is not helpful to the OP.