Well, there are more models of the 40W laser than there are units, so this may not apply to yours but mine will fire up the laser no problem without the computer. It has two buttons, one toggle and one temporary. The toggle one is enable/disable so you can do a "dry run" (run the program without powering the laser). The temporary push button overrides computer control and turns on the laser. The power is adjusted with a potentiometer.
Even if the mirrors are misaligned you can test the tube itself by looking at it (wear UV blocking glasses). Open the cover and when you manually power the tube via the push button, the tube should have a visible light beam in the middle along the entire length. The beam should be about 10 mm wide and light pink/violet in color, definitely not red/orange). If you put a piece of paper at the business end of it and fire a short pulse, it should burn a hole in the paper. I keep my laser at below 15mA and at that power it takes maybe a tenth of a second for the unfocused beam to punch through the paper.
If that works fine you should check mirror alignment. Start with the first one (right next to the tube and work your way towards the lens. The first mirror should reflect the beam at the middle of the second mirror and the spot should stay in the middle no matter where you move the Y axis. X axis is not important at this time.
When you have that aligned, check the second mirror (on one on the Y axis). Again it should reflect to the middle of the next mirror and the spot shouldn't move regardless of where the X axis is.
When I was first aligning them my beam was hitting the edge of the mirror and I thought it was fine, but overall result wasn't really good. I went back and redid the alignment which made a big improvement on the overall cutting capability. I also used that opportunity to replace the mirrors, since the ones that arrived with the machine had some discoloration and I was installing a new lens anyway. Clean mirrors are incredibly important and even a tiny speck of dust can have dramatic influence on the performance.
For alignment I use paper for thermal printers, they seem to give a better idea where the beam is hitting, but regular office paper will do.