"whatever kinds" is correct and usable in written English.
You can use it to indicate an unspecified or unknown number of types. For example: "The store offers whatever kinds of snacks you could ever want."
"And they want to be able to create whatever kinds of security and stability they can.
4. Line the pittas with a few pieces of whatever kinds of veg your kids like, then stuff with the tuna mixture.
Ask Your Car GASOLINE brands may be mysterious, but the difference in grades of gasoline -- between regular and whatever kinds of premium a station offers -- is straightforward.
Internet traffic is broken into data packets before being sent from one computer to another, and a packet sniffer can monitor the traffic, sniffing for whatever kinds of data it has been programmed to sense.
Because all Internet traffic is broken into data packets of uniform size before being zipped from one computer to another, a packet sniffer can sit passively on the network and watch the traffic go by, sniffing for whatever kinds of data it has been programmed to sense.
(March 25), in which William D. Brouse, the president of the Sport Utility Vehicle Owners Association of America, defended S.U.V.'s by saying that Americans have a right to drive whatever kinds of vehicles they want: Mr. Brouse says guns are like S.U.V.'s in that they pose no danger if used responsibly.
In straining for achievement, of whatever kind.
I could get whatever kind of dog I wanted now.
Of whatever kind you so desire, spiritual, physical, emotional.
Whatever kind of weapon it turns out to be.
Like ending a sentence with a preposition, or whatever kind of word "for" is.
