You should only use REPAIR_ALLOW_DATA_LOSS as a last resort.

Do you mean you haven't taken full backups? Have you taken diff or log backups?

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I think the the only you can do it is to separately install Express and then attach the mdf and ldf files to your Express default location.

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SQL Server will use all the memory it is able to, that is normal behaviour. What have you set as the Max Server Memory?

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Do you take log backups? If you have your full backup from yesterday then restore your logs up to the point where the data files were deleted.

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You should be fine to shrink the log on your principal as a one off activity, obviously it will be mirrored.

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Your log will probably be 60gb for a reason. Have a look at any maintenance tasks you have scheduled particularly index rebuilds as this is pretty heavy on the log....