When you apply the acidic cleaner and let dwell about 10 minutes, do you actually try and wash the brick or do more of a rinsining? With a good acidic cleaner, and a 25 degree tip a few inches off the brick, you should be having that come right off. You will have to be careful in the mortar areas, but usually you will have to sorta, "get on it" to remove the salts. Not to say it won't re-apperar with the irrigation but try some pressure and hot water and it really should go well. The sealants can help aid removal of the salts if they don't move the sprinkler heads. If there is moisture pushing anything through the wall, and the sealer used is permeable, then the sealer will not stop the return as it will push through the sealer. It really doesn't look like it is coming through the brick. The patteren is clear that sprinklers cause it and it would most likely be in more places if it is coming though the brick from a moisture source. You can check to see where moisture would be coming from, but it looks to be a wall and not have dirt behind it or some other source of moisture.