Hi Teleostomi,
I'm in over my head here on infinitive tenses, if any there be. I'm going to try to look it up. Perhaps someone else will come by in the meantime.
But I have a feeling you may be right about the present perfect. The missing cockroach would have been in the past, while the time of the illness is unspecified. (Of course we're talking about the making of the illness.)
Thanks for your confidence, although it may be misplaced. - A.
Edit. So far I find "infinitive is tenseless" and "infinitive tenses in Latin!" Your example must have another analysis.
Okay, I found : If we wish to speak of something relating to a time prior to that
indicated in the past tense we must use the perfect tense of the
infinitive; as, "He appeared to have seen better days."
I believe if you say, "He appears to have seen better days," the infinitive phrase does not change. It simply means the action the phrase describes took place before the time of the main verb. So I think your instincts were good in suggesting that the same phrase could be used in both a present perfect and a past perfect situation.
It sounds like the cockroach would have been expected to hang aroung awhile. Otherwise you could say, "There was no cockroach to make her ill." But that could mean she didn't get ill. The illness seems to be implied by the perfect infinitive.