If you ignore a lot of little pesky details, about 23.6 months.
LightSail 2 started at an orbital height of 720 km, so its orbital velocity was approximately 7.49 km/s. (Courtesy of Casio Computer Co.'s calculator)
Escape velocity is approximately 11.1 km/s, and LightSail 2’s speed increase is 0.1525 km/s per month.
But in reality, the answer is “never.” Per the LightSail FAQ:
LightSail 2 will then begin swinging its solar sails into and away from the Sun’s rays as it circles the Earth, giving the spacecraft enough thrust to raise its orbit (technically, the orbit semi-major axis) by several hundred meters per day. This portion of the mission will last 1 month.
LightSail 2’s attitude control system does not have the precision to maintain a circular orbit. Therefore, as one side of the spacecraft’s orbit rises, the other side will dip lower, until atmospheric drag overcomes the forces of solar sailing, ending the primary mission. The spacecraft will remain in orbit up to a year before succumbing to destructive reentry.
Though LightSail 2 cannot raise its orbit indefinitely, this would be possible by angling the sail more precisely during each orbit.
