This simple tutorial gives very clear instructions for making up a pillow / cushion.
Making up your needlepoint kit using this method, but using your kit dimensions as a template (rather than the pillow inner) is a good way to make up your kit into a pillow.
The tutorial uses a machine to stitch the pillow, but I used to stitch up all my needlepoint projects, big and small by hand and this worked fine.
If you have access to a machine for the full size cushion / pillow kits it will certainly be quicker, but it really isn’t essential to use a machine. Making up your mini kit into a cushion / pillow by hand is quick and easy anyway – no need for a machine for this little sweetie.