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I'm sorry to hear about your phone and the picture trapped in the internal storage. Unfortunately, as I understand, it's not possible to use and MHL (for HDMI) and USB OTG cable (for connection a mouse) at the same time.
I suggest that you turn to your local Xperia support team to see if you can get help at a service center to get your phone repaired without deleting the internal storage. Please follow the link below to find the contact information for your local Xperia care team: