You don't need a permit to day hike Mt Langley, but note that if you 'day' hike it will be an all day hike.
There is a recent trip report or other post in here of a party hiking Mt Langley and they did not need an ice axe, and certainly not over New Army Pass.
There is an alternate route up Mt Langley via Tuttle Creek, although it is seldom travelled. I have only read about it. It has a short class 4 section.
If Horseshoe Meadow Rd is closed, perhaps consider day hiking Mt Tyndall via Shepherd Pass up the 395, or go for a walk-up overnight pass.
