Our community had the zechus of hosting the Nikolsberger Rebbe for Shabbos and in his tish he dealt with this question. He answered using a yesod found in the Bnei Yisaschar. Chazal tell us that someone who is sick should ask a chacham (or tzadik) to daven on his behalf. Why should the sick person go specifically to a chacham? Tefilah is something everyone can do -- the choleh can ask a relative to daven for him, he can ask a friend to daven for him, he can daven for himself! The Bn”Y answers that sickness and suffering are caused by a person’s sins. When a person comes to the chacham, the chacham doesn’t just listen to the person's story – the chacham has empathy and feels that individual's pain. The chacham suffers along with the person. When that happens, it arouses mercy in Shamayim. The chacham does not deserve to suffer; he has done nothing wrong and is not guilty of sin. In Shamayim they are forced to lift the decree on the sick person and allow him to recover so that the chacham does not suffer undeserved pain.
Avraham, who had risked his life once already to rescue Lot, would surely have been sick and despondent had Lot not been spared from Sdom. Lot's pain was Avraham's pain. Therefore, the malach sent to heal Avraham and see to his well being could only accomplish that mission by saving Lot as well. This was not a seperate mission, but was part and parcel of the same task.
The obvious lesson here (which the Rebbe elaborated further on) is the need to feel empathy for another's plight, whether it be physical pain or emotional pain, whether it be a need for a job or a shidduch or some other problem.