、。
No matter how hard she tries, she probably won't be able to please him.
Literal
She [topic-は] however effort-do [even-ても], him [object-を] make-happy [the act-こと] [subject-が] cannot-do [probably-だろう].
喜ばせる is the causative form of 喜ぶ ('be glad, rejoice') — 'make X glad / please X.' The causative auxiliary ~せる/させる lets a Japanese verb express 'make/let someone do.' Here, the causative folds 'pleasing him' into a single action — pleasing as making him happy. ことはできないだろう ('probably cannot') stacks the inability ことができない with the conjectural だろう, expressing uncertain pessimism. どんなに~ても sets up the futile-effort frame: even maximum effort yields no success.