Switching from the Airprint driver to the CUPS driver seems to have done the trick! It's February 22 and I have been successfully printing without a single problem. That includes waking the printer from sleep, after computer restarts/shutdowns and powering off/on the printer. I've printed using a variety of apps on two Macs. I haven't had this kind of success printing since I bought the Brother printer late last year.
The fix is to select the Brother CUPS driver instead of the Airprint version when adding the printer to your printers and drivers in System Preferences. I had to do it two different ways on my Mac Mini and MacBook Pro. Both Macs are running OS 10.10.5 but I don't see any reason why this won't work with other operating systems.
Mac Mini
1. Turn off the printer.
2. Delete the printer from your printer list.
3. Turn on the printer and add it to your printers. (If it doesn't show up close the Printers and Scanners window and then reopen it. It should appear in the drop-down menu when you click the "+" button.)
4. You may see a window telling you that the printer software is not available. Click the blue button.
5. The next window will say "Add" at the top. Select the Brother printer if it isn't already highlighted.
6. At the bottom of the Add window click the "Use" button and select Brother (printer name) series CUPS.
7. Click the "Add" button.
On my MacBook Pro Steps 4-7 did not occur. I had to add the printer 3 times before the CUPS driver was automatically added instead of the Airprint driver. I have no idea why that happened. And one would think that the Airprint driver is the one to choose for wireless printing, but apparently not. There can be some mystery/magic involved when dealing with printer drivers and Macs.
Good luck!