This might sound like an odd thing to check, but see if the stupid IR sensor on the front is seated right; or even better, remove it.
When it's not mounted or improperly mounted the system will boot up and give the same "Error B" as it does when things are wrong elsewhere involving the keyboard...
...but if the sensor unit is burned out/shorted or not in all the way, it can often make the system do exactly what you are describing. It powers on, video output starts by blanking the display, but that's as far as it gets. The Junior I picked up had this problem as when I bought it the owner said it was delivering error B, by the time it was shipped to me it was going to a blank screen and no further until I opened it up and saw the IR unit half-in half-out. Removing it and reseating it properly was the only repair needed. (well, other than replacing the banjaxed floppy drive).
It's silly, minor, but something to check; and just another of those corners cut on the Jr's design that reminds me a lot of the "wishful thinking" in designs like the Apple II or worse, Apple III -- where things like expansion cards are held in not by latches or screws, but by the edge connector and wishful thinking.
A LOT of times all you need to do with systems of said age that are having problems like this is to just take it all apart and put it all back together. One wrong voltage, one unseated connector, one chip only half plugged in, and kiss off the whole setup.
See a XT clone I picked up recently that was shipped to me "working" and by the time it arrived half the DRAM had fallen out and were floating around loose inside the case.