After attempting this many times using the attack drone and turret super I switched it up a little and beat the boss almost immediately.
I would always die from some stray enemy so I used the sweep drone and turret super. The first few iterations are self explanatory. When you get to the four clones that spawn around the boss destroy the bottom right first. Then move to the bottom left and destroy that one. Head straight up, using a bomb when the green enemies spawn and destroy the top left. Then move in on the last clone from a distance, destroying the pink enemies before they get to you.
When you get to the last segment where the boss is chasing you and blue diamonds are spawning, you want to have at least one bomb and both of your supers left. The first iteration of the boss chasing you has blue enemies spawning on the borders and coming at you. Be patient and destroy the blues while you chip away at the boss's health until the next enemies spawn.
The next iteration will have yellow circles spawning that move east/west or north/south, but never both at the same time. When you get to this spot you are almost done. When the boss's shields go down use your turret super as close as possible to the boss. Watch out for the rotating walls. If you place it correctly the boss's health will plummet and you can pick off the remaining health with your guns.
The final iteration has blue and yellow enemies spawning with the walls growing longer and thicker around the boss. Again wait for the shields to go down and drop a turret close to it. It's health will plummet again. All you have to do is pick off the last little bit of health and you're in the clear.
It took me about a half dozen attempts to defeat the boss this way. I know you give up the firepower, but I enjoyed the extra protection. Electrostatic's solution is sound, I just kept getting killed by a stray enemy and I had to try something else after 50 tries or so. I tried not to use my last bomb until the final iteration. It can get a little hairy if you misplace the turret and having that bomb may give you enough time to take it out.