Is the melodic error (laḥn) that occurred in the Takbir not invalidate the prayer, so I should disregard it due to obsessive thoughts, or is it obligatory to repeat the ‘Asr prayer that I already repeated, or must I repeat all the prayers of the day, and what is required of my wife who prayed behind me?

It is clear that you are afflicted with severe obsessive whispers (waswasah) regarding the pronunciation of the Takbir. The cure for this is to completely ignore these whispers. If you doubt the correctness of pronouncing a letter from the letters of the Takbir, do not repeat it. Most likely, your doubt about pronouncing the letter "kaf" correctly is not real and is due to the influence of waswasah. Even if we concede the validity of what you mentioned, it would be beyond your ability, and you would be excused for your inability to do it. Therefore, the prayers are valid, and there is no need to repeat them, or to instruct your wife to repeat them.