There is the new Eka Hotel. Drive time for these three is about 15/min to/from. When booking to ask if they provide free transfers.
These hotels can be expensive, so many choose to go farther into the City for stay at The Fairview or their lower cost sister hotel (on the same ground) The Country Lodge. If you choose these, taxi is about $20 each way.
Assume next morning's flight to JRO is earliest about 8am, so at that hour and since check-in for an international flight is 2/hrs, there won't be much traffic.
But know, having to overnight in Nairobi, hubby will have to pass thru Immigration, purchase Kenyan Transit Visa USD$20 and must have a Yellow Fever inoculation and proof of same for entry to Tanzania.