Well, technically, you've only told us the time to travel across
our galaxy, so we don't have enough information to calculate
anything about its length.
Fortunately, some of us are vaguely aware that our galaxy is
roughly round, at least in its plane. So "across" means its diameter,
and that's the same in any direction through the center.
Relieved, we can now proceed to calculate:
(3 x 10⁸ meter/sec) x (8.64 x 10⁴ sec/day)
x (3.65 x 10² day/yr) x (10⁵ years)
= (3 x 8.64 x 3.65 x 10¹⁹) meters
= 94.608 x 10¹⁹ meters
= 946,080,000,000,000,000,000 meters
(about 587,867,000,000,000,000,000 miles) .