How do I repair a brick?

That depends on the extent of the erosion, but if it's not that bad you might want to consider other options that mortar. You might be able to seal a few cracks with a special type of ing. I forget the name, but it's used for furnaces and it's fairly heat resistant. We did that once with our fireplace, but we didn't burn wood in it. We just had a gas space heater.