SQL services wont start due to incorrect drive letter after windows restore
I assume you've restored entire partitions, but the partition drive letters are now wrong.

If SQL is looking for I: then you have three choices

a) Fix the drive letter of the partition that's now got the data.  You said this is not an option.  Is that because I: is already taken, or because something else is now looking for this data based on its new drive letter (E:)

b) Add an additional drive letter for the partition - nothing stops you referring to the same partition by more than one drive letter.  This isn't an option if I: is already taken and you can't move that data from I:

c) Create a symbolic link on your new I: drive to point things back to what's now on E:.  Start an elevated command prompt.  Then type

I:
cd "\Program Files\"
mklink /d "Microsoft SQL Server" "E:\Program Files\Microsoft SQL Server"


If you already have a Microsoft SQL Server folder on your new I: then change the mklink command to make the MSSQL.1 folder as the symlink rather than the Microsoft SQL Server folder.