Your problem may be that the definition of Nash Equilibrium might not overlap with your intuition of what 'equilibrium' should be. A NE is just a strategy prof

Your problem may be that the definition of Nash Equilibrium might not overlap with your intuition of what 'equilibrium' should be. A NE is just a strategy profile for every player of the game such that no player would want to change her strategy profile if she knew the strategy profile of all other players. If the second player loses to the first player's strategy profile whatever she does, that is still an equilibrium because she wouldn't gain from changing her action to something else that also guarantees loss.