To assemble reactors you need to drag a reactor into the bottom center slot in the engineering bay, same as with mechs and weapons. To ignite a reactor each slot on it must be filled.

When you assemble the reactor, once you added piston and injector, you should be able to click the yellow button on the right side.
First click open the button "protection cap", and second click actually push the button.
Then the reactor start spinning, and the temperature gauge indicator move (to show it is effectively started)

After that, you can grab the reactor (there is a handle shaped part right in the bottom middle of the screen, just above the reactor small icon on the UI) and place it with other reactor on the reactor list on the left.
You'll see ignited reactor differ from shut down ones as the ignited one is brigther.

Then you can finally open the mech tab, put a mech on the assembly bench, and you should be able to put the ignited reactor in it.

I know its late jaja but I'm figuring out how to play this game too. You have to drag the reactor and drop it in the box at the center bottom of the screen. In my case, there was a mech being displayed in that thing. So I figured out that THAT thing is the construction table.

So, that's it, drag the reactor into it and it will show you the reactor assembly section