You can't just throw a wire over the balcony with an internal tuner like
that. That sounds like an end-wire which those tuners won't handle. I
don't think the 570 has the range of some of the earlier Kenwoods like the
850.
I'd suggest that you try getting a real tuner, you could save money in the
long run. You can find some decent tuners used. You'll need a way to read
SWr. You can use a VCO hooked across the meter if you don't have an
accessible watt meter.
Apartment antennas are a significant compromise, no matter what you have, so
don't expect to win DX pile-ups. But, there are some stealthier ways you
can get things to work.