No Video or Bad Video.
Even though a Vidiot board gets 10 flashes on the diagnostic LED, it can still have problems and not work. A common failure is to have absolutely no video output at all. Assuming the 13" monitor is working, the first thing to suspect is the TMS9928 chip at U16, as this 40 pin chip is very problematic. If this chip is having problems, often the video will be distorted or won't sync properly. But you will usually have some kind of video.
The TMS9928 chip outputs the video signals directly to two video amplifiers, LM359N at U18 and U17, and then goes to the output video plug at J4. If either or both LM359N chips are bad, you will either get absolutely no video, or video that won't sync. These video amplifier chips do fail (just repaired a Vidiot board where the TMS9928 was fine, but both U17 and U18 LM359N chips were bad).
Also note the video signal passes thru a number of resistors around the LM359N video amplifiers. If the colors are strange or won't sync, check these resistors. In particular there are three 100 ohm 1/2 watt resistors than often go open (R89, R97, R102). You will have to remove one end of the resistor from the board to test with a DMM. Also the three 3.9k resistors (R101, R96, R88) mounted just above the 100 ohm resistors should be checked too.
The pots for the colors and sync also go bad often. The green and sync are 10k ohms, and the red and blue are 2.5k ohm pots. Replacement is a good idea.
Video ground is another problem. There is a large ground plane on the front and back side of the Vidiot board. Make sure these two ground planes "buzz out" and have continuity with each other (a small jumper may be required to ensure this). Also if the video is over-saturated with blue or red, there may be a problem with the video signal ground at J4 pin 2. The factory often has a capacitor going from Vidiot J4 pin 2 to ground. If red and/or blue are over-saturated, replace this resistor with a jumper from J4 pin 2 to ground. This should fix the saturation problem.
Also check the two 74LS374 chips at U1 and U2 as these are problematic too, though this may not reflect directly in the video signal.