the model name/number would be on the outside of the retail display box.
i've always had great luck with logitech myself. my old keyboard that came with the mx1000 keyboard had a range of over 50 feet through walls.
a relative recently got one in the $60-100 range and it seems to work fine at greater than wired keyboard distances. never checked the maximum range on that one though.
if you own an exact copy, not just one that looks similar then perhaps there is something wrong with the keyboard. if its a replica try the old keyboard in the same ports to see if it works. if it does then i'd just return the bad one and try again. if you get another sucker then i'd get a different model. hard to say its hit or miss when you only got one bad one.
i think i spent about $200 on my original mouse and keyboard set. it was one of the first laser mice made by logitech. not sure if it was bluetooth or wireless.. maybe that has something to do with the range. looks like the current gen mx series set (bluetooth) is only around $130 or so.