Probably you can't do a decent job in Lightroom. This is really a task for Photoshop or Photoshop Elements.
It is possible to use the Spot Removal tool to copy the open eye and move it into place, but as you've realised, you can't mirror flip it in Lightroom.
As DJ says - it's a job for a pixel editor like Photoshop.
PS is the best application to use for this purpose, but it is possible to do this inside LR with a few extra steps.
1) Inside the Develop module create a Virtual copy of the image and select menu Photo> Flip Horizontal.
2) Inside the Print module select Layout Style> Single Image/Contact Sheet.
3) Select both the image file and the flipped Virtual copy so they appear on the contact sheet.
4) Under Print Job'> Print to: = JPEG file, File Resolution = 600*, JPEG Quality = 80
5) Under Color Management> Proifle = sRGB, Intent = Perceptual
(*File Resolution needs further discussion.)
Click on 'Print to File button, select the folder where the original image file is located, enter the original file name with the number 2 added (File Name 2), and click Save.
You'll need to Synchronize the folder for the new image file to appear in the Library module. Use the Clone or Heal tool to fix the closed eye. You could also do this using another picture that has the person with eyes open, instead of a flipped Virtual copy. Once completed use the Crop tool to isolate the corrected image.
If you have more than a few images that need this type correction Photoshop is a much better choice. Here's an example: