We claim the digits are $\overline{555555 \dots 6}$ which works by considering $34^2 = 1156, 334^2 = 111556, 3334^2 = 11115556, \dots$. Then note that $33^2 = 1089, 332^2 = 110889, 3332^2 = 11108889, \dots$ and $35^2 = 1225, 112225, 1112225$. This can be formally proven by defining $t = \frac{10^n - 1}{3}$ and considering \[ (t + \varepsilon)^2 = \frac{(10^{2n} - 10^n) - (1 + 6\varepsilon) \cdot 10^n + (1 - 6\varepsilon)}{9} + \varepsilon^2 \]for $\varepsilon = 0, 1, 2$.