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Mix the mortar in a bucket according to the manufacturer's instructions. We used a buff-color mortar mixed with some sand.
Load up a large grout bag with the mortar and doublecheck the consistency to make sure it will flow at a controllable rate.
Apply the mortar to the two column sections on either side of the alcove first. Be sure to work in sections so the mortar doesn't change color in the middle of a run.
After all of the joints in the columns are filled, come back and dress them using a masonry joint tool. The mortar needs to set up about halfway to a slightly crumbling state. In our case it took about 40 minutes. It's not necessary to make the joints perfectly neat since the brick veneer surface is rough.
After the mortar sets up, use the same techniques on the inside of the alcove.
Next, move on to the front wall and the herringbone pattern.
This process can be long and tedious -- be careful and patient, and try not to get mortar on the brick faces as it can be hard to clean.