You can't really test components when they are in the circuit. It's also difficult to tell is the relay failed because, although you can hear the relay switching, it doesn't work then it might be whatever is controlling the relay (which is the rest of the board). You could remove the relay from the board and test it though. It's big and easy to work with and you have a big part number on it so you can find a replacement. If it's any other component that is not working, you're probably SOL.
You could go for a hail mary and measure the resistance across the large electrolytic capacitor with the circuit unpowered and see if it reads a short-circuit (5-10 ohms or less). If it does, it probably means one of the chip-sized decoupling capacitors on the board as failed. Since they usually fail short-circuit since they are all in parallel. These capacitors would be most of the ones that look like look like the vertical row of components on the left side of the PCB as well as the one large orange block in the middle-bottom of the PCB between the 16 and 8 pin ICs. The large electrolytic would look puffy or be leaking if it was damaged.
The problem is that since they are all in parallel, the only way to find the bad one is to remove them one-by-one until you run into the bad one.