Another option for removing debris is to limit how much of it you want to keep around. One of the settings allows you to choose anywhere from no debris, to unlimited debris.
This option is available under Settings in the main menu (the menu you see when first starting the game, not the in-game menu), and is called Max Persistent Debris.
Keep in mind this setting only handles debris from not controlled flights; it won't automatically kill everything that gets detached from your current launch. That only happens when you end the flight, or switch to the Space Center.
Not quite what you were asking for, but the only other option, yes, is to intercept each and every piece, and manually de-orbit it.
The part most interesting for you is:
lat = -0.0708559885838229
lon = -97.2394889192516
alt = 2917405.7211654
To have a stable geosynchronous orbit, you need to set latitude to 0, so you are over equator. To obtain longitude, simply launch another craft, and keep it in KSC. Read lon from it's save. Proper altitude can be read from Kerbin entry at wiki. Currently it's 2,868.75 km - you are a bit too high, so update accordingly.
Now to the orbit part, you want:
ECC = 0
for exactly round orbit. Also, to avoid confusing your game, you need to set average of periapsis and apoapsis to your new-current altitude. However, when we take into account Kerbin radius, it gets 3 468.75 km, due to Kerbin's ~600km radius:
SMA = 3468750
To be sure you are on equator, nullify inclination:
INC = 0
The tricky part is MNA - Mean anomaly at epoch. If your game will decide to use it instead of lat and long set above, you may find yourself not where you wanted. Experiment with changing it a bit in one way or another.
For easier experiments always edit quicksave. That way you will be able to load updated state with one key. And always backup your persistence file.
For a video tutorial watch this Youtube video. As usual, Scott Manley is your friend.
Best Answer
If you would like to do it without mods, you can use the Whack A Kerbal in the debug menu. Open the Debug Menu by pressing Alt+F12. Then select the "Show Whack A Kerbal" option. Fire the object by pressing the Middle Mouse Button. I would recommend quicksaving before doing this, in case you miss the intended target.