Sick-person [topic-は] nurse [by-に] attentively care [object-を] has-been-done.
手厚く ('attentively, generously') is the adverbial form of 手厚い. 世話をされてきた combines passive (される) with てくる in past tense — 'has been receiving care (over time, up to now).' The passive frames the nurse's action from the patient's perspective.