It sounds, to me, like you are physically shifting the shift lever, but not actually advancing the shift mechanism.
That pic of the shift lever shows it took a hit. You may have bent something in the shifting mechanism or worse, in the trans.
Put the bike on it's center stand or somehow prop the rear tire off the ground.
With the motor off, put the bike in first gear. You should be able to shift through the gears while spinning the rear tire. The transmission needs to be spinning to properly engage the dogs on the gears.
Are you able to actually get to fifth gear?