Is my arduino board broken?
I assume that PWR LED is turned on?
If on USB connection PWR LED is off and on 9V connection it turns on you should check fuse. It is on the right side of USB connector and is protection only from USB voltage source.

If it works you should check level of voltages or try connect usbasp and then using avrdude check if it is connecting normally.

Few ideas for voltage levels: 


Voltage on "main" ATMega328 (You can check it through ISP header)
Voltage on Arduino's chip which is ATmega16, or CH340 if you have chinese one (or something like that). It should be the same as on main Arduino, but to be sure...
Voltage on reset


You need around 5 volts to be sure that it works properly because of construction of arduino's board.

If you still have no answers, no voltages you should check Arduino's schematic for more information: https://www.arduino.cc/en/uploads/Main/Arduino_Uno_Rev3-schematic.pdf

If there is electrically everything all right and you did not check with usbasp (or other AVR-compatible programmer) I believe that next step is to check bootloader on your board.