Playing with Taylor series is not helpful enough. What else would you try out?
$$\sum_{n=1}^{\infty} \frac{\arctan(1/n) H_n}{n}$$ $$\approx 2.1496160413898356727147400526167103602143301206321$$ It's easy to see the series converges since $\arctan(1/n) \approx 1/n$ when $n$ large. Maybe its integral representation makes us feel more comfortable
$$1/4\int_0^1 \frac{ 2(\gamma \pi x \coth (\pi x)+\gamma) +i x \left(\psi ^{(0)}(-i x)^2-\psi ^{(0)}(i x)^2-\psi ^{(1)}(-i x)+\psi ^{(1)}(i x)\right)}{ x^2} \, dx$$