Do I understand it correctly, that your idea is to check by trial division every odd integer "n" in the interval 10^104<=n<10^105 whether it divides N?
Except you don't have to check every odd integer. You only have to check against the primes.
Also, actually you don't have to start at 10^104 and go all the way up to 10^105, you only need to start at:
245246644900278211976517663573088018467026787678332759743414451715061600830038587216952208399332071549104
and go to the square root of the RSA number. Since you already know that both numbers have to be 105 digits, this number outputs a quotient right at the limit of breaking 105 digits. All you have to do is know every prime number in the space between that number and the square root and then systematically divide it out.
It's still probably an impossibility with all these handicaps in place, but nobody has said that I can't do it. I've got the primes, all I lack is a program to systematically divide them. My programming skills have been going down the drain as of late. If anyone wants to help with that part, I'm up for any help that can be given.