OP - you indicated that you tried different aperture and speed settings to no avail.
The K1000 uses a 'galvanometer' in its metering circuit - this is basically an electromagnet with two electrical sides. If the electrical currents from two photoresistors are balanced (equal), then the needle will go to the middle position (proper exposure settings). If the needle pegs at '+' or '-' that doesn't change with the lighting level, then there is probably a short in one circuit or an open wire in the other. Alternatively, the meter might be fine, but the camera may be experienceing an overexposure situation. You indicate that you pointed the camera at a bright light; have you tried pointing towards a dark area?
The K1000 has a battery check function that will also 'peg' the needle at the top '+' part of the exposure scale. Set the camera to ISO 100, speed on 'B'. A good battery will peg the needle.
Your LR44 battery is suitable for the K1000.
If you are comfortable troubleshooting basic electrical circuits (following a schematic, using a ohmeter, soldering, etc) then you might be able to fix it. The K1000 service manual is available online, and there are many discussion threads in various forums on K1000 meter problems.
Here's a link to a short discussion in another forum with several ideas:
The Classic Camera Repair Forum: K1000 Light Meter
- Craig