Reduce Turing Machine
I don't want to spoil the exercise, but I'd try to play with the fact that the property

$$
p_M(n,k):\ \mbox{$M$ accepts $n$ in exactly $k$ steps}
$$

is decidable. Then, we have that $\langle M \rangle \in T_{n+1}$ iff there are exactly $n+1$ pairs $(n,k)$ satisfying $p_M$.

Try to exploit this in your reduction.




  Can I use some kind of search algorithm to find a string in it and exclude it?


This approach is unlikely to work: to search, the new TM $N$ would need to run $M$ on some given inputs, but this is risky since $M$ can diverge and "block" the whole execution of $N$. It's safer to work with decidable predicates like $p_M$.