Inside an LCD TV, there are effectively just a few major components:
- LCD matrix
- Power supply
- Backlight driver (aka, high voltage power supply)
- Main board
- Video board
- T-Con board
- IR Receiver
Some of these items may be combined with others (control board and video board might be the same) or not used (backlight driver usually only needed for CCFL backlights).
Let's work through that list and assess the probability of each component causing the problem you have described.
Your LCD matrix is probably fine. A failure here probably wouldn't stop the TV from otherwise powering on. A failed backlight probably wouldn't either.
A bad power supply might work enough to turn on the red light you mentioned in a comment, but fail to respond to a power-on command. A failed power supply is a likely candidate.
Your TV does not have a high voltage power supply.
The main board is responsible for receiving your power-on command and controlling the power-on sequence. A failed main board is also a likely candidate.
Your TV does not have a separate video board.
A failed T-Con board is not likely to cause your problem. And I don't think your TV has one.
There is a small chance that a shorted speaker is triggering a short-circuit protection mechanism, causing the TV to immediately power off. I would consider this unlikely.
When the TV does work, you state you are able to turn it on using the on-board buttons and the remote, and when it doesn't work neither of them work. Therefore we can rule out the button board and the IR received.
There is a small chance that if you open up the TV and remove the power supply board, you might be able to identify a failed component and replace it. More likely you will have to replace the entire board.