Hi, and Welcome.
My first suggestion is for your map site, just in case you don't know, to use green markers (laavu=lean-to) or red (kota=kota, duh!) for sleeping, orange are places you can make fire at but not necessarily for sleeping. Blue markers labeled
autiotupa are small cottages free to sleep in,
päivätupa are intended for short stops, not sleeping and
varaustupa need to be rented. Grey markers are either sights or bird watching locations. Sorry for rambling if you already knew this.
Second,
this map might be usefull for planning your trip if you want to find more "natural" places.
Their main site at
http://www.nationalparks.fi/, particularly the
self-guided tours section for ready routes is worth checking out, but I'm guessing
Päijänne is the kind of place you are looking for?
Lahti would make good starting point (I am in no way biased just because I live here
) since it is in the southernmost bay (it's name literally means "bay") of Päijänne region with straight train lines from Helsinki. Lahti-Vääksy-Heinola is popular water route there and south-east of Heinola river Kymi connects city of Kouvola to Kotka by the coast, but I'm not sure how accessible that route is since there are few dams, might have to carry your kayak.
Right next door is
Saimaa, largest lake in Finland and probably another place you want to visit, with
Savonlinna worth a special mention.
Saimaa and Päijänne together make up the so-called Lake Finland (or "Lakeland" as some PR guys want to name it) region filled with lakes and rivers, many of them connected to either Saimaa or Päijänne (this seems to be present in URW as well, I've noticed), so it's probably the place you want to focus on. There's plenty of islands off the coast as well, especially near Helsinki, Turku and Kotka, but I don't know how kayak would handle that.
But that's all popular commonly visited places.
Hossa is the most recent National Park and doesn't have much in the way of turist crap yet (
aside from that one steel pontoon), and from the map it seems to connect to Oulujärvi lake, where river Oulujoki runs to city of Oulu at the coast. That route would take you pretty much across the country, east to west. That might be your best bet at "wilderness" (unless you want to go all the way into Lapland, Lake Inari would be worth visiting), since most river and lakesides are populated to some degree, summer cottages at the least.
Cheers, and remember not to drink and paddle.