There are generally two methods used to make the connections to the glass of the LCD.
1. A rubber strip divided into very small conducting and insulating sections squashed between the LCD and the board.
2. A clear (usually) flexible PCB strip which is attached by conductive glue to the LCD and the board.
I don't know details of how these are produced/attached but I have tried to repair such with no success (40 years servicing electronic things).
In the U.S. there is a store that gives away meters like that for free, called Harbor Freight.
That doesn't help you, but my point is that such a meter is very inexpensive. The battery inside is worth more than the meter. I would not invest time in fixing it, but instead spend time finding another one.