Hope this isn't too late for your trip - There are things to do in Brunei, but Mulu caves, just oiver the border in Sarawak are so fabulous I would say try to go there too. I have lived in Brunei for 5 years, driven all over Sarawak and Sabah and still haven't finished!
Go to Temburong Canopy walk - as long as your knees are up to climbing over 1000 steps! Mine arent unfortunately, so haev never done it. Everyone who ever has has come back raving about it. There is now an (expensive new ) hotel in the jungle heard $300 per night - unconfirmed ... Day trips from 7am till dusk half that cost per person. Speedboat through mangroves in Brunei Bay, short road trip, upriver in a longboat, jungle climb and canopy, cool off in ice cold clean river, longhouse visit and food.
Kampong Ayer (watervillage) tour - very interesting contrasts of wooden house looking like death (not inside though!) and opulent Royal palace on skyline.
Evening tour to see proboscis monkeys - well worth it
Empire Hotel and Country Club - enjoy a bit of how the other half lives without having to pay to stay ...great "afternoon tea " in the Foyer - fabulous five story atrium in gold leaf and italian marble
Royal Regalia Museum - see all the golden gifts given to the Sultan over the years - including the golden car used for Royal weddings ....
Mulu Caves.- Tour can be arranged through a Brunei travel agent, or you can start in Miri and move into Brunei, but I wouldn't miss Mulu if I was you. Stunning caves - three different groups of them. Fly in for a two nights three day stay in the Royal Mulu Hotel . Built of wood up off the ground in amongst the trees. Lovely staff - the warmest reception of all hotels I have stayed in. Casual, but of a reasonable standard. Was a couple of years ago I went in and I believe the prices have risen ...don't they all?... But i think the walks, the guide, the caves were enough to warrant it. I intend to do it again now my son is older and probably more able to appreciate it better (moaned all the way walking through the hot jungle! ).
Clearwater caves stunning swimming spot at the base - be ready to dive in, be shocked by the cold water, but not want to get out!
Short walk into Lambir Hills national park (1/2 hr out of Miri) to lovely waterfall for a picnic - great swimming hole for a lazy day or walk further over the hills to others