Use a long tool or stick to wedge the brake pedal, or have a friend depress.

Get a volt meter, and check the thin circuit board at the brake light bulb socket/holder. You should have 12 volts.

If you do, then sand paper the terminals on the light bulb socket, bend them slightly to give you a little more pressure. You can very lightly sand the circuit board where the light bulb socket terminals contact. Also try and seat the socket just a fraction of an inch less than where it is now, sometimes the terminal leave a dimple.

If you dont have twelve volts at on the circuit board trace where the light bulb socket terminals land, then remove the white connector the mates to the circuit board. Follow the traces from the light bulb to the flap that acts as a "male" connector. Match these to the sockets on the white connector and check if you have twelve volts on the white connector. If you have voltage on the connector, then try and very light sand the flap that forms the male connector. Mate the white connector to the board, and see if light goes on. If no lighht, try jiggling the connector. If still no, use your meter to check connectivity from the flap to where the bulb socket mates. If you dont have connectivity, you might need to order a new circuit board.

If you don't have voltage on the connector, repost.

Its detective work, but it's almost certainly cured by a piece a fine sand paper.

When you've got the light on. Apply a little dielectric grease to all the connection. If you dont have a volt meter, you still go through all the steps of light sand paper, but go to radio shack and buy a cheap one, you dont need an expensive meter just to do this.

And use very fine sand paper with a light hand, the trace on the board is thin, you just taking off the tarnish/corrosion.