Let me start this by saying that it is easy to theorise on others' headache, but shake all heavens with crying when we have it. What I want to say is that on the face of it your problem may appear as very simple and routine happening to many others. So please do not rush and do anything harsh and irreconcilable.
It is just second time in three years that this overstay of your wife at her maternal house happens. For all normal married people's view, this will not appear as any serious issue, but as 'Chalta hai" in every family.
In the present question case, I am not sure what made you feel that there is some rupture and you need some harsh steps. What was the issue and duration of the previous instance? How it was resolved? How is your day-to-day life and relationship?
Relationships do not break in one day. They will be showing warning signals and there will be enough time to repair or remedy the issues. In your case also, I think there could have been such mild and gradually aggravating signals and issues. If so, did you find the base reasons and causes? Did you take any positive remedying and reconciling steps then and there? How did things deteriorate to such level that communication between you both is broken and you have brought mediators? What is it in your place or with you that makes her uncomfortable?
Without knowing those things, it is not possible to give a suitable suggestion.
Taking the issue narrated here as a standalone one, I feel that she has some reason. You know this well.
Problems between husband and wife will only worsen if third party is brought in between. The best way is to communicate directly. It may be explosive and provocative initially and at times. But both, especially you, have to maintain cool. After all, it is your family life that is at stake.
Hence I suggest that:
1. Immediately leave the thought of sending legally worded or formal letter. Instead send a one-to-one frank and private letter to your wife, following the points given next.
2. First try to talk with her on phone. Forget and leave your prestige and ego. After all you are going to talk to your wife and going to avoid your life going bad. Even if she does not answer, try for a few times. Then send a Whatsapp voice message telling that you want to talk to her and tried, but could not get. Let the language be polite and even apologetic. Nothing to lose. Convey that you really miss her.
3. She has lost her mother. So she emotionally very much down. Tell her that you understand that and want to be with her to support her in this situation. Tell her that you will try to go over there if not for Covid lockdown.
4. Sincerely try ways to go to your in-laws house where your wife is now. Convey to her your sincere efforts on this regard and also update her. Even if she is not replying, go on sending Whatsapp message/voice message. But ensure that all your messages and communications are sincere, simple and plain and never harassing or pressurising.
5. Never involve third parties. Never get provoked .Take extra care to be cool and patient. Do not report everything to the third party including the 'so-called friend'.
6. Keep away from the 'problem matter' as far as possible. Make this know to your wife.
7. Three or four years is too short a time for husband and wife to grow into mature partners. You both are still in the early fragile period. So only sincere intention to forgive and forget and mutually accommodate can solve the problems and keep the relationship going.
8. Just forget and remove the idea of sending a letter but sincerely ask her to come back at the earliest as you really miss her. Treat and consider her as your young wife and not as an enemy. There will not be any need o a letter for which this question is asked.
(Note: the question is not clear as to whether it is for you or you are asking for your friend. Hence my suggestion may be suitably taken as addressed to the relevant person)