My mother sent me a letter and asked about my health.
Literal
Mother [topic-は] letter [object-を] gave-[conjunctive-くれ], my health [object-を] asked
くれ here is the 連用形 (conjunctive stem) of くれる, linking two actions without て: gave a letter and asked. This literary linking form avoids the casual feel of てくれて. 尋ねる = to ask/inquire (more formal than 聞く).