The traditional way to tackle the Around the Mountain walk at Taranaki is to start and end at East Egmont, taking four days to do the circuit; this leaves enough time to include the summit walk on the last day, if you've got enough energy left. Here's the route I took:
|1||East Egmont to Lake Dive||-|
|2||Lake Dive to Waiaua Gorge||-|
|3||Waiaua Gorge to Holly Hut||-|
|4||Holly Hut to East Egmont via the Summit||-|
I'm afraid I don't have the distances to hand as the maps aren't available to me right now, but there's plenty of documentation available from the Department of Conservation (DOC).