Without spending a fortune on a kayak ergo there are still useful exercises. These will tend to use body weight and and/or minimal equipment.
A lot of the effort in kayaking comes from the core, so a good all round core routine will be an important part of any land-based training. Canoe and kayak magazine published a set of core exercises for paddlers a few years ago (part 1, part 2).
You also need a few strength exercises, such as bench rows (only needing a bench and a dumbbell). With access to a gym you may be able to do pulley rows; a pulley machine is also good for the very important rotator cuff. Or there may be a lats and traps machine. This is designed to be used with a chest pad; as a kayaker I prefer to use it without, but with lighter weight, and work the relevant bits of my lower back. Don't neglect your legs completely - they're an important part of your connection to the boat.
There are various guides online here's one that concentrates on body weight exercises. There are also several books dedicated to the topic.
Try to maintain or even improve your flexibility; at least stretch properly after a strength session.
General fitness of course has to be maintained by whatever means you like. Apart from usually being open all year round, swimming has the extra advantage that it's a necessary skill for water sports so worth practicing. It's also less biased to legs than running or cycling.
This is very close to being an all round strength and fitness programme, so if you're into training in company a circuits/core class plus some cardio and a few shoulder exercises should be very useful (but I have neither the time nor the personality for gym classes so can't be more specific).