Look in your yellow pages phone book under "Janitorial Equipment & Supplies" and phone around to the places listed there to find out who sells S. C. Johnson Wax Co. products in your area.
The S. C. Johnson Wax Co. is the largest manufacturer of floor finishes in North America, and they are well respected in the janitorial service sector of the economy both because they make good products and because they train their sales reps exceedingly well. Anyone selling S. C. Johnson Wax products in your area will know the local sales rep, and if they're not sure what product would best suit your needs, they'll contact him for advice.
Tell them your situation and explain that a primary concern is that any finish you put over the floor must be:
a) easy to maintain without a floor machine. Tell them you need something that will allow you to just scrub any dirty traffic areas down with an abrasive nylon scouring pad to scrub off the dirty surface layer and then just put a new coat of finish down.
b) won't be slippery when wet (if you don't have a door mat in front of that door), and
c) any other requirements you can think of.
I don't know of any way to get the existing tiles smooth again, but you can restore a gloss to the floor by using a very hard sealer over it.
Aside:
You need to understand something about floor finishes though. Floor finishes are softer than sealers because they're meant to be maintained by periodically scrubbing off the dirt embedded surface layer and recoating with another coat of floor finish. Unless you want to start spending a lot of money on a floor machine (mine cost me $1700 $Cdn), then you'd probably be better putting on a much harder sealer (instead of a finish) which won't get embedded with it's too hard for the dirt to become embedded in. In that case, if the floor sealer wears off or does get dirty, you can clean the surface with a strong acid (to dissolve the embedded dirt) or with an abrasive nylon pad (or even some fine sandpaper) to remove the dirty surface layer and put down another coat of sealer to restore the gloss and protection. A floor sealer meant for vinyl composition tile floors is made primarily to protect the floor from stains. Sealers are not meant to be removed, and will dry much harder than finishes. However, you can remove a sealer with the appropriate strippers and equipment. You might need to hire a janitorial service company to remove sealer from your floor, but you could do it by yourself if you wanted to. It would take you a while though, but you could do it.
