The letter（主语） she looked forward to（定语从句,宾语是省去的关系代词that或者 which) came （谓语）at last （状语）
coming 是不能作谓语的!
全句意为： 她盼着的那封信终于到了能不能举几个look forward to doing 的句子？We look forward to seeing you again.我们期待着与你再次相见。I look forward to coming to work every day.　 我每天都迫不及待来上班。We look forward to working with you to achieve that. 让我们同心协力，实现这个目标。We look forward to serving you again, sir. 我们期待能再为您服务。