。
I asked him to reconcile them.
Literal
I [topic-は] them [object-を] reconcile-[causative-させる]-[for-me-てくれる] [in-the-way-ように] him [to-に] asked.
Complex causative-benefactive stack: 和解させる (causative of 和解する 'reconcile' = 'make reconcile') + ~てくれる (do for me) + ~ように (asking that) + 頼む (ask). The sentence says 'I asked him to make them reconcile, as a favor to me.' A rich multi-auxiliary construction.