。
She has been grieving his death for nearly ten years.
Literal
She [topic-は] ten-years nearly [emphasis-も] his death [object-を] is-lamenting-grieving.
十年近くも stacks three elements: 十年 (ten years), 近く (nearly), and the emphatic も (highlighting the surprising duration — 'as long as that'). 嘆き悲しむ is a compound verb joining 嘆く (to lament) with 悲しむ (to grieve), intensifying the sorrow beyond what either verb conveys alone. The ~ている marks this grief as an ongoing state stretching across nearly a decade.