I can't sleep after my mother's death 2 months ago. I have outburst of crying during the day and night. Will I ever sleep again?

You lost your mom and you're grieving and mourning that loss.
Yes, you will be able to sleep again. What might help you through this sad and hard time is a bereavement support group. Also, ask your doctor to prescribe a mild, low dose, anti-anxiety medication like lorazepam that you can take as needed. That will help too.

You're still grieving. It's only been 2 months, so be kind to yourself. Grief is not a one and done. It takes time. And everyone is different in how they handle their grief.
Talking to a counselor or therapist can certainly be helpful, as can joining the support group Grief Share. Most of their meetings are on Zoom right now. You can also try taking some Melatonin before bed, and see if that won't help you sleep better.
My husband died a little over 9 months ago, and I still have moments where I cry. I'm not sure those moments will ever go away completely, but I do believe in time they will be fewer and farther between. That is my prayer for you.
So give yourself time to grieve, and know that what you're experiencing is quite normal. God bless you.

Please seek counseling to help you process your grief. I lost my mother over 10 years ago and still think of her often, but now more often with smiles than tears. Losing a beloved mom is a special kind of pain. I’m sorry for your loss and hope you’ll seek help and find peace soon

Perhaps you can Google “GriefShare” and find a local group.

I’ve heard great things about this program.

I'm so sorry. It really does get better over time but that doesn't help with the sleep now. Do you think talking to a grief counselor would help?