There was a narration from the Holy Prophet (Sallallahu Alaihi wa Sallam) that once a companion of his married and informed the Prophet that he had married a widow. The Prophet asked him why he married a widow. He told him that if he married a single woman then he would have found her being playful towards him and he would have been playful with her.
Although I don't have the reference for the above narration, nonetheless, I need guidance from you for a similar case.
Two of my paternal uncles married two women who were sisters in relation to each other. The younger uncle divorced the lady after a year of marriage. They couldn't have children because of a severe illness of the girl and consequently separated thus severing relationship between our family and my aunt's (the elder one) family.
The divorced girl was the youngest sister of my aunt and has a very good and chaste character. She has recovered from her illness completely. Because of the fault of my younger uncle, much problems have been created in the family. I want to marry the girl because I believe that she will be a good wife since she is a good Muslim and is of my own age group (23 years). Moreover, I believe the marriage will strengthen the broken family ties.
Please advise me in my situation because the narration that I previously mentioned makes me believe that I need an Islamic scholar's advice on this issue.

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

The Hadeeth you mentioned is a sound Hadeeth reported by Al-Bukhari and Muslim which the Prophet said to Jaabir Ibn ‘Abdullah when he told him that he married a divorced woman.

As for your marriage to the woman whom you mentioned in the question if she is pious and of good character, then trust in Allah and propose to marry her. Her having been married before should not be an obstacle to such a decision above all if such a marriage will turn back severed kinship relations. Know also that all the wives of the Prophet were previously married before he married them except ‘Aa’ishah .

Even Khadeedjah his most beloved wife, was married before him. If you ask about Jaabir's Hadeeth, it is said that in the situation where one has the choice between two women who are both pious, both are of a good character, and both are beautiful, but one of them is virgin, the other was married before, no doubt that in this case it is better to marry the virgin one. But if the divorced woman is of better character; then she is better.

Allah knows best.