What length of tongue would be
: appropriate for a 17' kayak?

Example below is based on the tractor supply 1500 lb capacity 4x8 frame with 12" wheels. I replaced the 5' tongue tube with a 9' tube and added 3/4" ply floor and bunks for my 16 foot rowboat. The boat overhangs 2' in back.

Lets assume you want a maximum of 4' overhang on the back. (I think you must hang a flag on any extension 4' or greater and probably should anyway). Subtract 4' of overhang and 8' of trailer from your 17 foot boat. That leaves 5' of bow sticking forward of the bed of the trailer. Depending on your car, you probably want at least 2 feet between the hitch and the front of the boats for turning clearance and opening the trunk/tailgate. That pushes your tongue extension up to 7 or 8 feet. The tongue extends about a foot under the bed of the trailer so you want an 8 or 9 foot tube. That's an example. Adjust all of these numbers to fit your needs and measurements and round up for a little more flexibility. You can always shorten the tube if it's too long.

My trailer has the axle about a foot behind the center of the bed. With about 150 pounds of boat centered 2' forward of the center of the bed, my tongue load is around 80-90 lbs. That's a guesstimate, but it's a bit heavier than my son.

The whole rig is more flexible than it was with the stock tongue, but it is lightly loaded.

Thick wall tube, hitch safety chains, wiring extensions, etc. came from a local trailer fabrication shop. A long 9/16" drill bit and drill press help a lot when punching the holes in the tube to attach to the trailer frame. You can get by with a big hand drill for the 3/8" stuff to attach the coupler, chains, etc.

It might be worth explaining your needs and getting a quote for the trailer or tongue from your local trailer fab shop in the first place. The do-it-yourself option worked out better for me.