Start by finding out if it's the shifter or the hub. The fast way to do this is to put the bike on the work stand and pull the indicator spindle/chain until it maxes out. When that is totally pulled tight, you should certainly be in Low. If you can pull that indicator all the way out and get Low to engage properly, then you've got a shifter/cable/housing/pulley issue. I would replace the shifter with a decent Sturmey and make sure everything is slick - no catching for the cable anywhere on the pulley or in the housing. Before you do this be sure the indicator chain/spindle is screwed into the hub correctly so you know you're pulling it in correct position.
If you pull the indicator all the way and you still have problems engaging Low such that you're stuck in Normal, you've got a hub issue. It could be that the spacing for the clutch is off. This can happen because you've got a clutch spring that's too long or catching; an improperly seated thrust collar; or the low gear pawls are installed backwards/upside down. You have to get into the hub to fix these issues, but none are terribly hard to fix once you get in there.
Also be sure your drive side cone is adjusted correctly at all times when working on it: finger tight and back off 1/4 turn on the AW (1/2 turn or so on the FW or S5). Badly adjusted cones can throw off diagnosing it.