Another solution might be to get a cheap capture board and use the Svideo or HD output of the 360. Then you can just leave it full-time connected to the capture board, play in a window, maximize full-screen, take screenshots or record videos of your XBox sessions.
Keep in mind some have latency/lag when capturing, especially USB models... so gaming with 1/2 second lag isn't a very good option. A VIVO or PCI/hardware solution or similar should not. Also, by going VGA to SVideo or HD input may cause some signal degradation... but that depends on the resolution and output of the particular title/game.
There are many other solutions as well. Sometimes a new monitor is the fix- as some of the newer LCD's offer multiple inputs (like DVI + VGA) with a switch on the front panel for selecting inputs.
There are many, many alternatives. Higher end KVM switches also work nicely and as already mentioned, wont degrade signal much if they are good quality.