There is an exact sequence $$ 0 \to \oplus_n M_n(\mathbb C) \to A \to \mathcal K \to 0.$$ Thus, $A$ is nuclear as an extension of nuclear $C^*$-algebras, see vor example $IV.3.1.3$ in [Bruce Blackadar, Operator Algebras: Theory of $C^*$-Algebras and von Neumann Algebras].

Now, any nuclear embeds into $O_2$ by Kirchberg’s Embedding Theorem (which even applies to exact algebras), see [Eberhard Kirchberg, N. Christopher Phillips, Embedding of exact $C^∗$-algebras in the Cuntz algebra $O_2$, Journal für die reine und angewandte Mathematik 525 (2000), 17– 53.].

The algebra $O_2$ is well-known to unital, simple, and nuclear, see [Cuntz, Joachim Simple $C^*$-algebras generated by isometries. Comm. Math. Phys. 57 (1977), no. 2, 173–185. ].

There is an exact sequence $$ 0 \to \oplus_n M_n(\mathbb C) \to A \to \mathcal K \to 0.$$ Thus, $A$ is nuclear as an extension of nuclear $C^*$-algebras, see vor example $IV.3.1.3$ in [Bruce Blackadar, Operator Algebras: Theory of $C^*$-Algebras and von Neumann Algebras].

Now, any nuclear embeds into $O_2$ by Kirchberg’s Embedding Theorem (which even applies to exact algebras), see [Eberhard Kirchberg, N. Christopher Phillips, Embedding of exact $C^∗$-algebras in the Cuntz algebra $O_2$, Journal für die reine und angewandte Mathematik 525 (2000), 17– 53.].

The algebra $O_2$ is well-known to unital, simple, and nuclear, see [Joachim Cuntz, Simple $C^*$-algebras generated by isometries. Comm. Math. Phys. 57 (1977), no. 2, 173–185.].