I suspect the login window is present but you can't see it. For some reason your machine believes that there are two screens connected
Aug 2 05:10:47 iMac WindowServer[150]: Display 0x042735c0: Unit 0; Vendor 0x610 Model 0x9cd7 S/N 0 Dimensions 23.50 x 13.23; online enabled built-in, Bounds (0,0)[2560 x 1440], Rotation 0, Resolution 1
Aug 2 05:10:47 iMac WindowServer[150]: Display 0x003f003d: Unit 1; Vendor 0xffffffff Model 0xffffffff S/N -1 Dimensions 0.00 x 0.00; offline enabled, Bounds (3584,0)[1 x 1], Rotation 0, Resolution 1
The login window is probably showing on Unit 1, but it may be seen as offline?
Check the cable connections on your GFX card. If you have more than one cable connected disconnect all but the working one.
If you only have one cable, disconnect it, then re-connect it...
If the issue persists, try the following....
The blurry screen is normal in a multi-monitor setup, the second (not primary) screen is blurred until you log in.
Have you tried moving the mouse pointer horizontally. You will probably find that it will be constrained on three sides (top, bottom, and either left or right), but will move horizontally off/on the visible screen across one edge. That will give you an indication of where the "other" monitor is logically located.
Assuming you are at the login window, and that you know the user name and password: move the cursor off onto the "ghost" screen, click once, press the first letter of your usual log in name, press Enter, type your password, press Enter.
Hopefully that will log you in. Let's try and change your default desktop onto the screen you have working...
Press CTRL F2
Press the down arrow three times
Press Enter
That should launch System Preferences
press CMD F (also known as Apple F, also
F) to get to the search bar of the System Preferences
Type Displays
Press Enter
Hopefully you will now see the Displays preference panel on your screen
Click the Gather Windows button
You should now see both Displays preference panels, move them so they don't overlap
Now click the Arrangement button
Drag the white bar (representing the menu bar) from the current screen rectangle to the other screen rectangle
That should set your active display as your main desktop...