Ok that's got the RE5R05A box (the petrol car had the same box and even 2WD variants used the same box without the transfer case of course). Petrol engined vehicles had different torque converters and valve bodies.
Nice thing about that box is that you can dump the oil and refill it easily. It has a dipstick for easy checking of the levels but the ATF does expand a fair bit as temperature rises so having the gearbox warm (30C to 50C) is probably easier and make sure the level's in the 'cold' range. Tne engine has to be idling when checking.
Examine the oil on the dip stick. If you see any discolouration (ATF is usually red) you may have a problem. Metallic parts (probably not on the dipstick unless it's a major issue) mean something's worn inside.
Change the oil. There's a sump plug at the rear, catch ALL of the oil and examine it for debris. If there is any, the type of debris will tell you if it's a bearing, gear surface or brake material. An experienced AT mechanic ought to be able to determine this and then give you a diagnosis fairly quickly.
In fact, it might be easier to do the whole job in a mechanic's workshop, so that if it becomes necessary to replace the box it can be done there. It's not a simple job to replace the box, if that becomes necessary and in Australian dollars they're about $4K reconditioned (new is a little hard to find these days!).
Another piece of good news - if it's a busted valve body, they can be obtained for about $600AUD second hand, or replaced with an upgraded unit for around the $1400 mark. You may have someone in your area doing similar to Wholesale Automatics in Victoria. Recently a member from here had their valve body replaced and reported smoother shifting and more responsiveness. A user from the States replaced theirs with an upgraded valve body and reported faster, more concise shifting than standard.
Personally I'd take it to an auto transmission specialist and discuss it with them. They may take the car out for a drive and instantly know what has let go in the box and be able to give you a rough idea of what it's likely to cost. Home diagnosis is great, hands on is a fantastic learning experience, but with only text and pics to guide any of us in forum land, this sort of fault isn't easy to find unless we've experienced these symptoms before. Fortunately for me, my gearbox still shifts and operates like new (after 260,000km).
The other thing you might consider (especially if there's any hint of milkiness in the ATF) is removing the ATF oil feed from the radiator and connecting an external ATF cooler. Our cars (well, not mine any more, I've done this mod) have a transmission oil cooler in the radiator. This both cools the transmission fluid when the gearbox is working hard, and warms the fluid up on cold mornings. However, if the unit becomes damaged, it can cross-contaminate both the coolant system with ATF and the gearbox with coolant. That's a big job to fix, and if your gearbox is ok now, adding the cooler externally to the radiator will prevent that from happening. If there has been contamination it may explain the symptoms and require a fair bit of flushing of both ATF and coolant systems to remove the impurities.