Can Khula be performed by sending notice?

yes. She can initiate khula, and it will be final when the husband accepts it. But, while taking khula, the wife will have to part with dower (full or a part) or some property. You have one more better option of approaching the court for dissolution of marriage. In this, the wife can seek divorce on the grounds provided in Muslim dissolution of marriage act 1939. In your case the ground can be cruelty.

Khula is the right of a woman in Islam to seek a divorce or separation from her husband. More accurately, it is merely the right of a wife to seek a release from the marriage bond.

Khula’ means the separation of the wife in return for a payment; the husband takes the payment and lets his wife go, whether this payment is the mahr which he gave to her, or more or less than that.

The basic principle concerning this is the verse in which Allaah says (interpretation of the meaning):

“And it is not lawful for you (men) to take back (from your wives) any of your Mahr (bridal-money given by the husband to his wife at the time of marriage) which you have given them, except when both parties fear that they would be unable to keep the limits ordained by Allaah (e.g. to deal with each other on a fair basis). Then if you fear that they would not be able to keep the limits ordained by Allaah, then there is no sin on either of them if she gives back (the Mahr or a part of it) for her Al-Khul‘ (divorce)”
[al-Baqarah 2:229]

The evidence for that from the Sunnah is that the wife of Thaabit ibn Qays ibn Shammaas (may Allaah be pleased with him) came to the Prophet (peace and blessings of Allaah be upon him) and said, “O Messenger of Allaah, I do not find any fault with Thaabit ibn Qays in his character or his religious commitment, but I do not want to commit any act of kufr after becoming a Muslim.” The Prophet (peace and blessings of Allaah be upon him) said to her, “Will you give back his garden?” Because he had given her a garden as her mahr. She said, “Yes.” The Prophet (peace and blessings of Allaah be upon him) said to Thaabit: “Take back your garden, and divorce her.”

(Narrated by al-Bukhaari, 5273).

There are basically three ways for a wife can seek a divorce:

1. Ask her husband that she no longer wishes to remain his wife, and the husband accedes to her request and pronounces a divorce on her.

2. If the husband refuses to divorce her, she has the right to Initiate divorce proceedings in a Court, whereby the Judge will study the conditions and reasons for seeking divorce, and then summon the husband and give him an option: either he accedes to the wife's divorce request and pronounces divorce on her, or the Court will use its authority and divorce the couple.

3. If the husband does not pronounce the divorce and the Judge sees a valid reason for divorce, the Judge has the authority to divorce the couple.

A Muslim woman may petition a Qazi to grant her divorce - over ruling the husband's refusal. This authority of the Qazi is subject to certain criteria which differ amongst the jurisprudential schools (fiqh), and subsequent to attempting a reconciliation between the parties, failing which further arbitration to seek an amicable solution and voluntary proclamation of triple talaq by the husband.

Ultimately the Qazi has authority to grant the divorce subject to the wife fulfilling requirements to return the mahr and compensate the husband by reimbursing him for what he provided during marriage, unless the husband is willing to forgo it, (which the Qazi usually encourages) unless the woman's action or behaviour has been such not to warrant it.

After the divorce, the husband is responsible for the education and maintenance of the children. The iddah period (waiting time after a divorce) of a woman who seeks a khula, is one menstrual cycle or one month if she is post-menopause i.e. ceased menstruating. This is to ensure she is not pregnant. The iddah period also allows for reconciliation between the husband and wife. If the woman is pregnant, then the waiting period is until she gives birth. There is still the need for witnesses when seeking a khula as in a talaq.

In your case, Husband is not ready to give Talak but allowing you to seek Khula which means separation is easy with out going to any court. Sometimes in the masjid, a Qazi will give a document related to Khula, where Husband consent will be recorded along with witness or you can file a Suit in family court to get the divorce decree.

When the wife files a suit for dissolution of marriage, the court issues notice to the opposite party i.e. the husband and if he fails to appear after the due process of posting and publication the court can proceed with the case ex-parte and pass a decree. In the case where the husband or his representative appear he is required to file a written statement following which the court has to fix a date for pre-trial proceedings for reconciliation. If reconciliation fails and the case will be contested by the following procedure and finally court will pass a decree for dissolution of marriage.

You can initiate by sending legal notice but when husband itself forcing to do khula means he is ready to give his consent. Among both, Shias and Sunnis, Khula is irrevocable. Other requirements are the same as in khula and the wife must undergo the period of iddat and in both the divorce is essentially an act of the parties, and no intervention by the court is required if not contested.