Assuming the problem is not BIOS related (e.g. something as minor as "BIOS battery drained" or "Disk configuration changed - Press FX to continue"), then you can try
Scientific Linux. Note that on the page I linked to, it indicates various boot options--one of which is 'ssh' to start an ssh server. Also note that, even though the page is described as "LiveCD" there is a link on the left for "Create Live USB"
Of course, a Live USB requires that your machine's BIOS be configured to boot off USB before local drives.
Also, I have never used Scientific Linux, and won't comment on whether it is good/bad/indifferent.
Given that the boot options on that page are to be entered at boot time and you have no monitor, you will need to guess when to enter the options. You can do it, because I've done something similar in the past. It may help to boot another device
with a monitor so you can get an idea of how long you have to wait and what extra keys, if any, need to be pressed before you get to the boot options.
Otherwise, look into remastering the live USB image to start the ssh server by default of look for another live usb distribution that does so automatically.