This doesn't work very well.
What you have to realize is that the weapon has to decelerate just as much as it accelerated to get into orbit. This will take a great deal of fuel, so the final impact body will be quite small. The useful term here is the payload fraction. The Space Shuttle is good example. It takes a liftoff weight of 2,000,000 kg or so to put 130,000 kg into LEO, for a payload fraction of about 0.065. Apply that to a 1 ton (1,000 kg) "weather satellite", and you get an impacting body of about 65 kg. The situation is a bit clouded by the fact that you presumably don't want a multi-stage deorbiter, but it's clear that 65 kg is optimistic, not to say unrealistic.
At 10 g's, retro burn time is only about 81 seconds and begins 32 km before the vehicle would pass over the target if there were no burn, so this is happening pretty much out of view of most radars.
Furthermore, this simply produces a satellite which is (temporarily) essentially stationary and which then begins free-falling, and ignoring atmospheric effects will take about 157 seconds, with a final velocity of about 1500 m/sec. Of course, you can't actually ignore such effects, and descent will really take a good deal longer. In order to get an 8 km reentry velocity will require double the total thrust (8 km/sec to stop the vehicle, and another 8 km/sec vertical speed) - you can't combine the two if you want a vertical reentry. So the impacting body will be even smaller. Like, in the neighborhood of 4 kg. And it won't hit an anything like 8 km/sec - remember atmospheric drag effects. You're right, though, to think that there is simply nothing looking straight up over DC, so there will be very little warning.
You may well feel that 1500 m/sec (about Mach 15) is pretty unimpressive. Well, that's because it is. Unfortunately, you'll get maximum impact velocity, about 8 km/sec, if you think of the final orbit as being a very slight ellipse, with closest approach being 120 km less than the maximum. Unfortunately, this will produce a very shallow approach angle which is in principle easily detected, not to mention producing a whole lot of drag and slowing your impactor down trementously. Fortunately for your story, the approach angle can be from the southwest, and there aren't a lot of radars pointing in that direction.