With patience, don't rush it, check & double check your soldering, with a bit of practice it's not hard though i have to admit i really hate these white on blue displays
After nearly 50 years of wiring things... If I DON'T double Check things... they Will go wrong... Somehow when I get in a hurry or assume anything, I'm Wrong...
So... Double Checking is and has been a habit for many years... Somehow that old hack about "Never enough time to Do it Right, Always time to Do It Again..."
Always Fits.
Not at all bad for a first time PCB, Even "Tin Plated it" I see... I could Never Get that stuff to work right, probably old chemicals... didn't even want to solder well. One recommendation though, Don't use the Bakelite or paper PCB stock... The FR4 material (Glass Epoxy) isn't really that much more expensive and it is much more pleasant to work with. it Smells Terrible when you drill it or saw it. Burning Phenol's aren't really good for a consistent diet.
At least it was for the 35 years I made circuit boards. Really great first attempt. I kept mine for 30 years, lost it when my wife died... Lost a Lot.
Are the holes large enough to accept a .025" square pin. If not you may be able to use a 2x8 socket
The're quoted as .025 pins but the're not and may fit. Same size as headers on Arduino.
Try some different sizes of solid wire and find one that just fits. Measure thickness.
Here is what I do. If .025" square posts will work, I solder a 2x8 male header onto LCD and then use ribbon cable with connector to wherever. I have made a small pcb to transition from 2x8 to 1x16 so I can plug display to a solderless breadboard.
If .025 does not fit try Mill-Max, for a header that will. See Mouser, etc.