"Hornborgasjön" itself is a good place for all sorts of birds. I suppose you have visited the official site
Hornborgasjön. The "Naturum" on the east side is a good place for various birds.
Beaver can be found in the lake also. Elk or moose (älg in Swedish) is not so common around the lake nowadays but if you have a car you can take trips around the countryside, preferably at dusk or dawn. Look closely at the edge of fields, you might get lucky!
Other places for birding depends on how far you are willing to travel.