As you note, "at seeing how fond the cat was of me" is a prepositional phrase rather than a noun phrase. Prepositional phrases generally serve to specify the time, location, or manner of an action (complete with subject and verb). So there are two ways to fix this sentence. The first, as noted in Akaisteph7's answer,, is to turn the prepositional phrase into a noun phrase by removing the preposition; "seeing" then acts as a gerund:
Seeing how fond the cat was of me made me disgusted and annoyed.
The other way would be to provide a subject to the main verb of the sentence, so that the prepositional phrase can act on it normally:
At seeing how fond the cat was of me, I became disgusted and annoyed.
This usage sounds a little strange to my ear (I would probably use "on" or "upon" instead of "at" here); but the OED notes that "at" may be used to indicate coincidence in time as well as in space, and gives the example "Our men gave them a shout at parting."