Kitten pees on carpet still ..help!

We have tried all the tricks that have been suggested to train a kitten to
pee in litter box and he is still peeing on the carpet. Here is the
background and what we have done so far:

He is 3 months old. (had him since he was 8 wks......a loving purring
kitten...great personallity except litter box issue). He has a litter mate
sister who uses the box always.
When they are closed in the kitchen area, he will use the litter box always
(where we keep them at night for now).
When we let him out to explore the house.......(limited to living room for
now),
he will sometimes pee in a certain spot on the carpet.
We will pick him up, each time, as soon as we see, and put him in the litter
box (usually too late).
I have never seen him use the litter right after that.
We clean w/ enzymatic cleaner to remove smell each time on the carpet.
He still goes back to the same spot and pees on the carpet. (several
cycles of this)
We then covered the area so he does not have access and he will find a new
spot to pee in another part of the room on the carpet.
Again we cleaned and deorderized and put a 2nd litter box on the spot on the
carpet in the living room on the spot.
He will approach the 2nd litter box box and pee just outside of the litter
box on the carpet.
We cleaned again. This repeats for a few times.
We then started to spritz him with a water spray when he start to see him
pee.
He will run and then immediately pee on a new spot against the wall on the
carpet.
We have seen him use the 2nd litter box to poop in interestingly.
Again, never has accidents in the kitchen.

Any new suggestions? At this point we hate to let him out of the kitchen
area until see some inclination to pee in the litter box. We would like to
introduce them to the hose when we are confident that he wont pee on the
carpet.

Thank you for the advise