Do you even see the bouncing Roku logo when you power it on? You might try a hard reset. Find the recessed reset button and hold it for 30 seconds and see if you get a display back.
However, the Roku 2 is so old that it's certainly possible that it has just failed.
I doubt its a the problem with the adapter because the white light turns on every time I connect the power cable. I tried hard reset already by holding it down for 30 secs