。
I was relieved that my broken English got through somehow.
Literal
I [of-の] broken English [subject-が] somehow got-through [because-ので] relieved-was.
片言 ('broken, halting speech') captures the experience of speaking a language one barely knows. 通じる ('to be understood, to get through') takes the message or language as its subject. ほっとする is an onomatopoeic-derived verb meaning 'to feel relieved' — from ほっと ('with a sigh of relief'). A relatable travel/language-learning moment.