Jason Maguire without actually seeing it, I highly doubt that this is possible. First off there should not be any room in a `15” chassis to accommodate the boards from the 24”. Then there is the bigger issue. A 15 panel will most likely not even get recognized. You would need to compare the datasheets for the old 24” panel with the 15” panel to determine if the LVDS connectors are wired identical etc. Most LCD driver boards actually have an EEPROM solder to it which makes it communicate with the main board and the panel so it will most likely not communicate with the 15 panel.
Se if you can find another Insignia TV that is broken in other ways and harvest the LCD from that. You can also try and check the label on the back of the panel. There should be the make and model for the panel on it. Do an online search with the panel number and see if it is available.