I'm not convinced that poor lauter efficiency is the main cause of OP's low mash efficiency. There is a very high probability that OP had lower than normal conversion efficiency, what with ~50% wheat (which might have been marginally crushed) and only a 45 minute mash. Plus, OP used a new grain suppler (and crusher), so that makes crush adequacy a suspect.
Unfortunately, OP did not measure one needed parameter that would allow the separation of conversion efficiency and lauter efficiency from the mash efficiency (mash eff = conv eff * lauter eff.) The missing data value is the wort SG at the end of mash, prior any sparge water being added. Being able to separate these two efficiencies allows the brewer to know whether the mash or the lauter (or both) was the primary cause of low mash and brewhouse efficiency.
OP is using a Brewzilla AIO, which has a malt pipe. So, it's brew in a basket. Lautering is accomplished by lifting the basket out of the kettle and letting it drain. Sparging is usually done by pouring the sparge water over the grain bed while still raised above the level of the wort. There is no drain valve to throttle. Thus there is no way to control the draining rate of the sparged wort. You can only control the rate at which water is applied to the top of the grain bed.
And, slow sparging only matters when fly sparging. When batch sparging, all you have to do is add the sparge water, stir well enough to homogenize the wort concentration throughout the entire volume, and then drain as fast as the system will allow.
Brew on