Okay, there's going to be some confusion here due to your wording.
A halogen bulb is an incandescent bulb filled with halogen gas. What you're looking at there is an LED bulb, which has no relation whatsoever to the word halogen. The 4R7 seen in the picture is just the value of the surface mount resistor that number is printed on.
We don't have much to go on here; for one thing we'll need to know the input voltage being fed to the LED through the red and black wires. I'm guessing it doesn't plug directly into the wall; is there a power adapter in that line somewhere? If so, what's the output voltage from it? AC or DC? That'll at least give us a start at figuring out what can be used as a replacement.
As far as your existing board, I'm a little suspicious of the connector to the left of the LED; it looks a little burnt to me. That may just be from age and heat depending on how old it it, or it could indicate a problem. What does the back side of that little circuit board look like? Can you get a picture of that?
Do you have a voltmeter? If so, it might help if you can measure the voltage across the red and black wires, along with the voltage present at the two pins of the LED itself.
Let us know what you find.