"handle" is correct, but the word you are looking for is "infinitive" not "infinite". " In this case "I am glad to have you handling the problem" would also be possible, with very slightly different nuance,
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Khai TranWould something like "I was happy to have you handled the problem" be appropriate for past tense?That is incorrect. Infinitives fo not inflect, so it should be "handle" whatever the tense of the main verb.