Can I determine the original resistance of a burnt/failed resistor?
Three things you can try:


See if you can read the colour code (probably not)
Try a multimeter, but this won't work if the resistor is broken
See if you can get the (sub)circuit of the device / some chip around the resistor and find the resistor there


Otherwise, I don't think you can find it out.

Note that you shouldn't use a damaged resistor (or any damaged device or component): it might give unexpected results and the properties may change over time. You should only use these methods to find a good replacement for the damaged transistor.