Storing mirror database failover details to the respective server or on repository
The concept is to create an extended event session listening to the DATABASE_MIRRORING_STATE_CHANGE event. This post covers the mechanics of querying extended event info using TSQL on live server, though haven't used this approach.

Otherwise, you can create the extended events session via SSMS, and configure that session to write the output to a file (xel) which can either be opened by SSMS or by using a system function that can convert the data to xml. The XEL ("event file") itself is binary. easiest way IMO is just to configure some kind of script to copy those files to a central instance, then open and process there.

Fields that DATABASE_MIRRORING_STATE_CHANGE event captures:  https://docs.microsoft.com/en-us/sql/relational-databases/event-classes/database-mirroring-state-change-event-class.

See https://msdn.microsoft.com/en-us/library/hh213147.aspx  for step by step to set up the an extended events session that dumps results to an event file. (needs to be done per instance, though can be scripted). Fairly quick setup to see if it's giving you all the details you need.

You can include the DATABASE_MIRRORING_STATE_CHANGE event class as part of a trace as well.