Hi,
"There was no cockroach to have made her ill."
Is the tense of "have made her ill" in relation to which tense: now or at that time?
This seems like quite an ambiguous sentence . It's unclear whether or not she actually became ill although, if she did, it was not caused by a cockroach. It is also unclear as to whether she was still ill at the time of speaking.
In practice, the context would usually make all these matters very clear.
Best wishes, Clive