。
I'm happy I get to look after the baby.
Literal
Baby [attributive-の] care [object-を] can-tend [nominalizer-のが] happy.
世話を焼く is an idiomatic verb+object expression for 'to take care of, dote on, look after' — literally 'to burn care,' where 焼く is a fossilized idiomatic use unrelated to 'burn.' 焼ける is the potential form, 'can take care of.' The nominalizer の + subject particle が wraps the whole preceding clause into a nominal serving as the subject of 嬉しい — 'the fact that I can take care of the baby is happy-making.' English can't match that structure and has to unwind it with 'I'm happy that...'