How long to broil burgers if no meat thermemoter handy?
It really depends on their thickness, how hot your broiler gets and how you like them done.
If they are less than 1/2 inch thick, and you want them medium rare in the middle, I would probably do them for about 4-6 minutes a side, considering you are doing them frozen. I would at least set them out for an hour at room temp, which will start to defrost the outside, but not the middle. That will keep your centers cool enough to not get overcooked. But if you like them well done, they may start to burn on the outside before the middle is no longer pink. So a longer sit time may be necessary. And if they are brown enough, you may want to check the center of one, to see if it's cooked to your liking. If not, you can always turn off the broiler, but keep the pan in the oven to "bake" them a little longer in the residual heat. Put them on a lower rack from the broiler too, so they don't burn and become hockey pucks. Hope that helps!
As phurstluv says, there are a lot of factors involved here. Generally speaking, when you see liquid rising to the surface, it's time to turn, and when you see liquid rising to the surface again, it's time to rest. You can also use the poke test: a medium burger is about done when it feels like the flesh between your thumb and forefinger when you hold your ring finger against your thumb. If in doubt, make an extra test burger. You can always go back to the broiler if they are not quite ready.
It might also benefit you to try and defrost, if possible. Frozen stuff is impossible, and you might end up with a burnt puck with an internal temp of 70, especially in a broiler.
All that said, I would opine that 160 is way too hot a temp for a burger. Chicken, sure. But a burger, IMHO, should end up at 140 tops.