I have a problem with application of Mazur's Lemma.
Just consider $B(H)$ when $H=\ell_2$. Then, $B(H)$ is a normed vector space. Then, take operators $$X_n:={\rm diag}(0,0,\cdots,0,1,1,1,1,\cdots)$$ (the first $n$ values are $0$). Then, $X_n\rightarrow_{wo} 0$. But any element $Y_i$ in the convex hull of $\{x_i, x_{i+1}, \cdots ,x_n\}$ has the form of $${\rm diag}(0,\cdots, 0, \alpha_{i},\alpha_{i+1},\cdots, \alpha_n,1,1,1,1,\cdots).$$ I don't know why there is a sequence $(Y_n)_{n=1}^\infty$ that converges to $0$ in norm. (By Mazur's Lemma, there shall be such a sequence).
In Wikipedia, it uses ‘strong convergence’，in fact，it is just the norm converge in $B(H)$ which is different (and much stronger) from the strong operator topology in $B(H)$. If the 'strongly converge' in Wikipedia is just the usual strong operator topology, then it makes sense. I don't know where did I make mistakes.
The claim you are trying to prove is the following statement.
Proposition. Let $H$ be a separable Hilbert space and let $(T_n)_{n=1}^\infty$ be a sequence of operators in $B(H)$ which converges to some $T$ in the weak operator topology, i.e.,
$$\langle T_nx, f\rangle \to \langle Tx, f\rangle$$
for all $x,f\in H$. Then there is a sequence $(S_n)_{n=1}^\infty$ of convex combinations of elements of $(T_n)_{n=1}^\infty$ which converges to $T$ in the strong operator topology, i.e.,
$$\|S_n x - Tx\|\to 0$$ for all $x\in H$.
Proof. By the Banach–Steinhaus theorem, the set $\{T_n\colon n\in \mathbb{N}\}$ is norm-bounded, hence so is $C$, the convex hull of this set. We wish to show that SOT and WOT closures of $C$ coincide. However, a WOT closed set is SOT closed and $T$ is in the WOT closure of $C$, so we only need to show that $T$ is in the SOT closure of $C$. To this end, it is enough to show that for any finite set $\{a_1, \ldots, a_k\}$ in $H$ and $\varepsilon > 0$, we have $\|Sa_i - Ta_i\|<\varepsilon$ for some $S\in C$ and all $i$. However $(T_n \oplus \ldots \oplus T_n)_{n=1}^\infty$ converges to $T\oplus \ldots \oplus T$ in the weak operator topology of $B(H \oplus \ldots \oplus H)$, so
$$(Ta_1)\oplus \ldots \oplus (Ta_k)\in \overline{\{(T_j a_1)\oplus \ldots \oplus (T_ja_k)\colon j\in \mathbb{N}\}}^w. $$ Thus, by Mazur's lemma, $$(Ta_1)\oplus \ldots \oplus (Ta_k)\in \overline{\{(S a_1)\oplus \ldots \oplus (Sa_k)\colon S\in C\}}^{\|\cdot\|},$$ hence $$\|(Ta_1)\oplus \ldots \oplus (Ta_k) - (Sa_1)\oplus \ldots \oplus (Sa_k)\|<\varepsilon$$ for some $S\in C$.
So far we have not used that $H$ is separable (and actually the above reasoning works for any Banach space, not neccessarily only the Hilbert space $H$). This will be important now.
The unit ball of $B(H)$ is metrisable in SOT (hence so is the SOT closure of $C$), thus limit points in the SOT closure of $C$ are attainable as limits of SOT convergent sequences in $C$. $\square$