I've been fiddling with this idea for a while and have basically come to the conclusion direct VGA capture isn't worth the trouble. There are VGA to HDMI converters available for about $20-$30 that do the job surprisingly well. There used to be a number of VGA -> digital chips available (TI TVP7002, etc) but they've all been EOL'd or NRND (Not recommended for new designs.)

For HDMI/DVI capture at this scale, the best option seems to be an FPGA based system. There are several boards available that have the necessary hardware but are expensive (relative to a 1-port IP-KVM), power hungry, or large (or all of the above.) The Digilent Zybo may have enough to pull it off (and has the ARM cores to boot) but is still a bit on the pricey / power hungry side. I'm hoping the new Snickerdoodle Zynq based boards will be able to do the job (with an add-on board)

Regarding using the Pi2, it might work, especially if it can be fed via the camera interface. The lack of USB-OTG (linux gadget support for KB / Mouse / Mass Storage) limits the usefulness somewhat.

I've been hopeful about the Beaglebone Black / X15 due to the USB-OTG and the tons of extra IO.

The other board of interest is the Dragonboard 410C - it can supposedly capture at 1080p60 (more than enough for the average system you'll stick a KVM on) but I'm not sure what that means explicitly (yet.) It does have USB-OTG (though it can't do USB Host / Device simultaneously.)