Before you do anything, it's often cheaper to find a broken camera to get the part (E18, bad CCD or a broken LCD will work). Depending on the failure, you may be able to test the flash before opening the cameras to do the repair.
More often then not you can find a camera with an issue like an E18 or a lens that's bad but comes out just enough to avoid the E18 crash for next to nothing. As long as the camera works for parts, it will suffice for what you are using it for. In many cases, these are in as good (or better) condition then the camera they are being used to repair. Flash board failures are rare, relative to how many have bad boards and lenses.
However, I'd suggesting seeing how cheap a used camera that's just like this one that works 100% is. If you find the prices on the parts cameras aren't much lower then a used one that works, it might make more sense to buy a new one and use this one for parts.
If you attempt the repair, a lot of soldering is involved; you should also let the camera sit for a good week or so to avoid being shocked by the capacitor or frying the camera. If you have never soldered before, practice on junk boards first.
If you are still having issues, the flash from the donor may be bad or your camera has board issues. At this point, it's going to be cheaper to buy a new camera unless your donor camera has the part you need to repair yours. If you do swap boards, transfer the entire case so the serial numbers are consistent; it isn't required but it's a good idea.
Here is the guide to replace the flash board: Canon PowerShot A590 IS Flash Assembly Replacement