Hi Rob,
Lake Burley Griffin is the man made lake in the middle of town and the prime species here is carp. Whilst they are fun to catch, if you are looking to sample some of the regions trout fishing then you will probably be better served getting out of town a little.
The Cotter and Goodradigbee rivers are about within an hour of of town and may be worth a try. Alternatively, you could head out a little further an fish the Snowy Mountain area, plenty of smaller streams to give the tenkara a workout. There are also quite a few guides operating in the Snowy region which should make things a little more straightforward.
If you decide to spend a weekend out of town, I would suggest checking out Lake Crackenback resort – has all the nice things to keep your wife busy while you explore. Or perhaps something in the town of Jindabyne – another town on a Lake.
Good luck and enjoy your visit, Malcolm