He loves his fiancée but she wants to cancel the engagement; what should he do?

Firstly: 

What you have mentioned
about your relationship with your relative and the haraam actions that you
committed is something that is very regrettable when it happens among
Muslims. It is indicative of fathers’ and mothers’ shortcomings in
protecting their daughters and preventing them from mixing with men, and
being careless with regard to relatives concerning this matter. 

We praise Allaah for having
given you insight into your mistakes and enabling you to repent and regret
it and seek forgiveness. We ask Him, may He be glorified, to forgive you and
accept your repentance. 

Secondly: 

You should find out the
reasons why this girl wants to cancel the engagement. You could delegate
this task to one of your female relatives. There may be reasons that can be
remedied. 

If the girl insists on her
position, then all you can do is be patient and seek reward, and perhaps
that will be better  for you, because one does not know where good will be,
and with whom he will find happiness. 

You should also pray
istikhaarah, asking Allaah for guidance, and ask Allaah to help you to get
married to a righteous woman. 

Beware lest you be one of
those who are weak in faith, who – when they miss out on something they had
wished for – lose their strength and willpower, and indulge in haraam
things, or they fail to do obligatory duties, or they announce their
withdrawal from life. That is the attitude of those who are hopeless and
filled with despair. But the believer knows that everything happens by the
will and decree of Allaah, and that he will never get anything but that
which has been decreed for him. Hence Allaah says (interpretation of the
meaning): 

“No calamity befalls on
the earth or in yourselves but it is inscribed in the Book of Decrees
(Al‑Lawh Al‑Mahfooz) before We bring it into existence. Verily, that is easy
for Allaah.

23. In order that you
may not grieve at the things over that you fail to get, nor rejoice over
that which has been given to you. And Allaah likes not prideful boasters”

[al-Hadeed 57:22, 23]

We ask Allaah to guide you
to what is good for you and make it easy for you wherever it may be, and to
enable you to have a happy and successful marriage by His leave. 

And Allaah knows best.