I think the T595 Daytona only has a side stand so that won't be easy to do without an assistant  or a paddock stand, Christof.

I had the Speed Triple 1050 but I think the swinging arm and adjustment for the chain are broadly similar.  IE - single sided swing arm and a bloody great nut on the right hand side to hold everything in place.  The hub is mounted on an eccentric arrangement so that turning the adjuster clockwise loosens the chain and anti-clockwise tightens it.  There's a single pinch bolt that locks everything in place once the chain is correctly adjusted.  I think the correct tension is around 30 to 40 mm taken from the centre of the bottom rung of the chain.

The good thing about this arrangement is that it's simple to adjust the chain AND maintain alignment because the mechanism automatically keeps the chain true.

As the chain sounds like it's had a rough and unloved life I'd start off with a clean slate and buy a new chain and sprocket kit.  You should get the best quality o-ring type you can afford.  It'll be more expensive than the budget alternatives but with correct care will mean you won't have to do the job again as soon as you would with the cheaper alternatives.

The chain will come with a soft link that joins the two ends of the chain together and will need to be riveted in place with a special tool.  That will set you back around £30 or thereabouts. You'll also need an angle grinder to cut off the old chain - another £30 odd notes if you don't have one.

It's a fiddly job that can go wrong if you've not done it before (which I'm guessing you haven't) so you'd best get a spare softlink if case you screw up the first time.  The last thing you want is a chain flailing around by your leg and getting spat out the back at the head height of someone following you.

If you don't have the tools then I'd spend the money they'd cost on getting the local bike shop to replace the chain.  It'll be done properly and be cheaper than the tools needed.

If the existing chain is loose I'd have a look at the top of the swinging arm, under the chain guard near the front where chain passes on to the front sprocket.  Has it become so loose that it's rubbing?  The Speed Triple has a Nylon rubbing strip as a sacrificial warning / last ditch safety after that it starts cutting into the metal of the swingarm.  I don't know if the Daytona has this.  Look and see.

If you're getting handling problems it might be the rear wheel axle mounting nut wasn't done tight enough - it's supposed to be really tight - around 108 foot pounds.  If it's loose it might we causing the wheel to wobble.  If the rear bearings are shot that would also cause the rear wheel to wobble.

Vibration might not be anything to worry about.  My Speed triple had horrible handlebar buzz at 50 mph - so bad I had white-finger after a short time and that was well looked after.  Or it could be down to your chain being buggered.  :)

Don't just buy a chain though.  Get the sprockets too - otherwise the wear on the sprockets will quickly transfer to the new chain.  In any event you don't want to go through all that bother only doing half the job.  Do it right from the get go and you won't have to revisit it again for a  long while - just do routine maintenance.

                                                                                                                                                                             

Cant believe what I've read here.

If your bike has developed handling problems, take it to a bike shop to get it checked. You may find its something simple, or not. But you should NOT be riding it everywhere until you get it checked.

Without seeing the bike, no one can diagnose a fauklt through a forum.

Please, take it to a shop ASAP.