Lower bound on number of oracle calls for solving $n$ instances of the halting problem
The result can be found in


Beigel, R., Gasarch, W. I., Gill, J., and Owings, J.  Terse, superterse, and verbose sets.  Inform. and Comput. 103 (1993), no. 1, 68–85. 


Their proof in fact shows that your problem cannot be solved with fewer than $\log_2 n$ queries to any set $X$, let alone to the halting problem itself. For some notation, your problem is sometimes denoted $C_n^K$ or $C_n^{HALT}$ (depending on your favorite notation for the halting problem).

For this and many related results, see also:


This survey by Gasarch
The book Bounded Queries in Recursion Theory by Gasarch and Martin.