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We hope that this reaches you in the best of health and Imaan. Ameen.

If the conditions of Nikah are not fulfilled, continued love relationship between a male and female who are non-Mahram is not permissible.

In order for a nikaah to be valid, it needs to be conducted in front of two witnesses and there should be an agreement of a mahr (dowry) amount to be given to the bride at this time. Just by two people saying it to each other will not count towards an actual nikaah occurring.

If you have developed a love for a non-Mahram (as you have indicated), and marriage is not possible (such as one partner deceived after some time, parents do not agree, etc), then steps MUST be taken to cure the heart from this love and desires which cannot be fulfilled.

First, a person should disassociate, or detach, themselves from that person and get rid of anything that reminds you about that person (gifts, letter/communication, messages). Do not go close to any such thing which reminds you of that person. Guard your thoughts as well and keep yourself busy with a positive activity.

Developing a love for Allah Ta'ala is extremely important. The human heart is the center of emotions and feelings. It tells us what it desires and works with the mind to dictate good and evil actions. To safeguard our hearts and to become beloved to Allah, it is necessary that we turn our love only into the love of Allah, our Creator and Sustainer. Humans love someone based on beauty, excellence, perfectness in knowledge, etc. Allah Ta'ala has the perfection of such qualities, and to Him all love should be directed.

Learning a life of piety will also help you in this situation. Observe hijaab completely. protect your honor and the blessings Allah has bestowed upon you. Convert time into gold. TODAY is the day of action, rush to do good deeds. Repentance and asking forgiveness can clean up mistakes of the past, and good intentions lead us to a golden future inshaAllah.

It is the nature of mankind to be forgetful and to commit sins. However, Allah has given us the ability to correct out mistakes; we may turn to Allah and beg for his forgiveness. To bring ease into one's heart, one should make sincere tawbah/repentance for one's sins. One should immediately abstain from the sin, regret and wish one had never indulged in the sin, and to make a determined niyyah to never return to the sin.
Allah Ta'ala is surely all-Merciful and all-Forgiving and is always ready to accept the repentance of His slaves, inshaAllah.

At this time, our advice, along with the above, is to review your life pattern and check to see how many ways you can come closer to Allah. When one corrects their relationship with Allah, Allah Ta'ala will correct his relationship with the rest of creation inshaAllah.

For extra duas and recommended readings, please visit the IslamEasy website. If you are particularly feeling depressed and saddened in this situation, please read the following booklet by Shaykh Yunus Patel titled
'Cure for Depression': http://islameasy.org/pdf/Cure_for_Depression.pdf

For dua-e-istikhaara, you do not need to have the dua memorized by heart. You offer two rakah nafl of salatul istikhara, and in the first rakat it is preferable to recite surah Kafiroon after surah fatihah and in the second rakat it is preferable to recite surah Ikhlas. After you finish the salat, call upon Allah in His glorious names and attributes, recite durood, and then read (from a book or paper) Dua-e-istikhara.

May Allah Ta'ala put ease in all your affairs and relieve you of any hardships. Ameen.
 

Please review this article:

Cure for Pre-Marriage Love Relationship which does not lead to a Marriage
http://www.islameasy.org/Cure_PreRelations.php

and also listen to this presentation :

[English]  For the daughters of the Ummah - protecting the honor and dignity of woman
http://islameasy.org/bayan/4DaughtersOfUmmah.MP3

[URDU] For Every Muslim Sister, Youth and Parents
http://islameasy.org/bayan/Taleem/4_Every_Sister_n_Family.mp3

Note: The rulings giving herein are based on the religious rulings of the Islamic Law and do not have any implications on the Law of the Country. The rulings given hereunder are specifically based on the question posed and should be read in conjunction with the question. AskAlimah.com site bears no responsibility to any party who may or may not act on this answer. AskAlimah.com site being hereby exempted from loss or damage howsoever caused. This answers may not be used as evidence in any Court of Law without prior written consent of the web site.