×So you should time your first strike carefully with the spawn cycle.

×the one in the lower right corner.

×the two in the center below her.

×No matter what I do, a spawn of roaches gets stuck in the center behind inescapable force arrows. I tried going from lower left to lower right, but the upper left queen went into the upper entrance and her spawn always goes into the enclosure unless I go to the right (impossible) or get above the enclosure (likewise impossible). Going from bottom roaches to bottom left also didn't work. What is the trick?

×The roach queens don't see Beethro and don't spawn before he makes his first strike.

×That's what I have been doing. Here is a demo of my last time through. What am I doing wrong? Thanks for the help.

×What you're doing - going across the bottom from west to east, driving a queen into the north entrance - leaves you in a very tight spot in the northeast corner. There's nowhere you can stand on the east side to keep roaches from the north from getting trapped on the arrows.
I suggest approaching the room from the opposite direction.

×Run north then west, wait for the two bottom queens spawn, kill the NE queen, deal with the horde, .... get stuck

×-Face south
-Wait for turn 30 before the first kill
-Go after the south queens, turning SE to kill the second one
-Head for the northeast

This is the hard part. You've got roaches from three queens and you have to both survive and not let any roaches get stuck past the force arrows.

The key is to get to (31,13) with sword NE. Then back up to kill the roaches north of you, and then turn your sword E. The east roaches should line up next to you, and you can reach the top row of the force arrows just as the roaches to your northwest hit it.

From here, cleanup is simple: the roaches spawned in the east passage will take a path that goes past the force arrows, so just clear out the other queens first.