Since you have successfully realised and distributed to the beneficiaries as an Executrix, it would be ideal to report back  to the High Court showing the details of disbursements that have been made.

Yes, as an executor or administrator, you must report back to the High Court on the distribution of assets to beneficiaries

You have to submit report to registrar testamentary division of HC 

A person to whom a grant of Probate or Letters of Administration is made is required to file a full and true inventory of the property and credits of the estate and of all the debts owing by any person in Court within six months, or such further time as the Court may allow from the date of the grant and to render accounts within one year, or such further time as the Court may allow (section 317).

Ask your lawyer to do the needful 

As an executrix with a Letter of Administration and Will annexed, you must submit an inventory and account of the estate to the High Court per the Indian Succession Act, 1925. This includes:

1. 
   Inventory Submission within six months.
2. 
   Account Submission within one year.

How to Submit:

• 
  Prepare Documents: Include a detailed inventory and a full account of distributions.
• 
  Verify: Sign and, if needed, notarize documents.
• 
  Send by Courier: Address to the Registrar of the High Court (probate department).
• 
  Cover Letter: Mention case details and purpose.

Tip: Confirm the mailing address with the court beforehand to ensure proper delivery.

The court responsible for issuing the Letter of Administration is typically the district court or high court, depending on where the assets are located.

Therefore you are required to file your report before the court which granted the LOA.

Section 317 in The Indian Succession Act, 1925

317. Inventory and account.—

(1)An executor or administrator shall, within six months from the grant of probate or letters of administration, or within such further time as the Court which granted the probate or letters may appoint, exhibit in that Court an inventory containing a full and true estimate of all the property in possession, and all the credits, and also all the debts owing by any person to which the executor or administrator is entitled in that character; and shall in like manner, within one year from the grant or within such further time as the said Court may appoint, exhibit an account of the estate, showing the assets which have come to his hands and the manner in which they have been applied or disposed of.

You may prepare the report in the manner as prescribed in the act and submit it through your advocate if you are not able to visit court for this purpose.

No you are not obliged. Unlesss court has directed you too

It's the responsibility and liability of the person on whose name the LOA is decreed. You just hire an advocate, who will intimate the same to the LOA holder through a Legal Notice. Your work finishes there.

if you are an 'executrix' i am not able to understand how you got a LA with Will annexed

always if there is an executor named in the Will, then a probate is granted

generally nobody files such a report after being issued the grant [probate or LA with Will annexed]

however if you get any notice from the Court then you can comply with such notice and file the said inventory report in Court 

- If there is direction by the high Court to submit the report of distribution of assets , then you have to comply the order after submitting the same 

- You can send the proofs of the same to the Registry of High Court with a list of documents , however it is better to take services of the lawyer. 

- If there is no order like this , then not required to submit the said reports.