In 1994 my son and I used Mt Madness out of Seattle (Scott Fishers old company, now owned by someone else) We went up the Western Shira Plateau route which takes an extra day or two. But the slow pace gives you gradual altitude benefit. This route also has you camp in the inner crater(18,500'+) for the last night. Thus you get up at a reasonable hr (and not at 1 am,) eat breakfast and finish the summit climbing starting at 8 am. I took Diamox only because I knew I would regret it if I failed to summit, but my son did not take it.
I have been told that the weather is usually best the last part of August to mid September. We had 7 hikers, 31 porters, which included 3 cooks, a chief porter,2 guides and 1 armed guard the lst couple of days for protection from poachers (one did get an eland our lst day) and wild animals.
This is the cadillac of Mt trips and we actually found ourselves over trained in prep for Kili after climbing over 50 peaks over 14,000' in Colo, many of them several repeats. The porters essentially do everything for you, which of course you are paying for. Be sure to donate any clothing, shoes, boots etc after the trip is over as many of the porters hike in rubber tire sandals, thin well used clothes and are very appreciative of your extras. The lst day I picked the porter with the biggest feet and told him my boots and running shoes would be his after the trip. I wear size 13-14 and I suspect he was only an 11, but he jumped at the chance to be my porter.