Obviously, if Grandpa was born on February 29th in a leap year, he only had (in the strict sense) a birthday every fourth year. Indeed, if he was born before 1900, his next birthday after February 29, 1896, would only be on February 29, 1904, because the year 1900 was a common year. Only the latter scenario is possible, otherwise Grandpa would be at most 58 years old, Kati at most 15, and her brother at most the same age (if they are twins), which means the sum of their ages would be far below 111.

On a birthday, we typically mean a birth anniversary, so from the $n$-th birthday to the $(n+1)$-th birthday, most people are $n$ years old. People born on February 29th after 1900 are $4n$, $4n+1$, $4n+2$, or $4n+3$ years old, while those born on February 29th before 1900 are $4n+4$, $4n+5$, $4n+6$, or $4n+7$ years old (we did not consider people born before 1800 or after 2099). According to this, if we denote Kati's age by $x$, then in 1960, Grandpa had his $x$-th birthday, and in 1962, he was $4x+4+2$ years old. The age difference $k$ between Kati and her brother is at most 3 years, because if they were born between 1896 and 1904, the sum of their ages would already exceed 111. Thus, Kati's brother is $x-k$ years old, where $k=0$, 1, 2, or 3. Now,

$$
x + (4x + 6) + x - k = 6x - k = 111, \quad 6x = 111 - k
$$

so $111 - k$ must be divisible by 6. Thus, $k=3$, $x=18$, and Kati is 18 years old.

Kati's brother is 15 years old, and Grandpa is 78, meaning he was born on February 29, 1884. Thus, before 1900, he had 3 birthdays, and after 1900, he had 15 exact anniversaries of his birth, totaling 18. Kati was born in 1944 (at the earliest on March 1st), and her brother in 1947, but both before September, because by the time the problem appeared, they had to have completed their 18th and 15th years, respectively.
When asked about her age, Kati replied in a veiled manner: I am as old as the number of birthdays my grandfather has had so far. Taking my brother into account, the total number of our years is 111. I note that my brother and I were born between two of my grandfather's birthdays. - How old is Kati according to this?