Learn Finnish: Minun näköni heikentyy. - My eyesight is weakening.

It is derived from the adjective heikko (weak). By adding the inchoative suffix -ntyä, it forms the verb heikentyä, which means "to become weak" or "to weaken" on its own.

How would the verb change if something else was weakening my eyesight?

heikentyä is an intransitive verb, meaning the subject itself is changing (your eyesight becomes weak). If something external causes the weakening, you would use the transitive verb heikentää. For example, Sairaus heikentää näköäni (The illness weakens my eyesight).

Why does heikko lose a k when it becomes heikentyy?

Adding the suffix alters the syllable structure. The root changes to heiken-, making the second syllable closed (it ends in the consonant n). In Finnish, closed syllables trigger consonant gradation, so the strong double kk from heikko weakens to a single k.

Do I need both minun and the -ni suffix on näköni?

In standard written Finnish, yes. When using a genitive pronoun like minun, the noun must take the corresponding possessive suffix. In everyday spoken Finnish, however, it is very common to drop the suffix and just say mun näkö.