hermit lake shelter is about 2-3 miles in
You can hike mt Washington in a day no problem, just get an early start so you'll have plenty of daylight. And of course remember while the Lion's head winter route has less avalanche danger than the ravines it still does have risk.. some good reading is on this site
"when it starts to hurt you might be nearly halfway, now its probably a good idea take out those ropes and put your crampons on!"
-Tim: MWOBS donor, AMC member, and steadily but surely working on joining the AMC 48 in winter committee