The tools that can be used are DNE, DNI, augmentation, MT, MP, &I, &E, and CP - nothing else. According to the Tomassi Logic Textbook, this should take 11 lines. I get the following 10 line solution.
{1} 1. $P \to \lnot P$ [P]
{2} 2. $P$ [A for CP]
{3} 3. $P \to \lnot P$ [A for CP]
{2,3} 4. $\lnot P$ [MP 2,3]
{2} 5. ($P \to \lnot P) \to \lnot P$ [CP 3,4, discharge 3]
{2} 6. $\lnot \lnot P$ [DNI 2]
{2} 7. $\lnot (P \to \lnot P$) [MT 5,6]
{ } 8. $P \to \lnot (P \to \lnot P$) [CP 2,7 discharge 2]
{1} 9. $\lnot \lnot (P \to \lnot P$) [DNI 1]
{1} 10. $\lnot P$ [MT 8,9]
I'm probably being really dumb here, but I'm not seeing how to do this in 11 lines. What am I missing?
$$\begin{array}{l}\{1\}&1.&P \to \lnot P&[\mathsf P] \\ \{2\} &2. &P&[\textsf{A for CP}] \\ \{3\} &3. &P \to \lnot P& [\textsf{A for CP}] \\ \{2,3\} &4. &\lnot P& [\textsf{MP 2,3}] \\ \{2\} &5. &(P \to \lnot P) \to \lnot P& [\textsf{CP 3,4, discharge 3}] \\ \{2\} &6. &\lnot \lnot P& [\textsf{DNI 2}] \\ \{2\} &7. &\lnot (P \to \lnot P)& [\textsf{MT 5,6}] \\ \{ \} &8. &P \to \lnot (P \to \lnot P)& [\textsf{CP 2,7 discharge 2}] \\ \{1\} &9. &\lnot \lnot (P \to \lnot P)& [\textsf{DNI 1}] \\ \{1\} &10. &\lnot P& [\textsf{MT 8,9}] \end{array}$$
In a Fitch format that would be:
$$\def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline#2\end{array}}\fitch{~~1.~P\to\lnot P}{\fitch{~~2.~P}{\fitch{~~3.~P\to\lnot P}{~~4.~\lnot P}\\~~5.~(P\to\lnot P)\to\lnot P\\~~6.~\lnot\lnot P\\~~7.~\lnot(P\to\lnot P)}\\~~8.~P\to\lnot(P\to\lnot P)\\~~9.~\lnot\lnot(P\to\lnot P)\\10.~\lnot P}$$
So, yes, those ten lines are valid, and you have used only the allowed rules — in fact only CP, MP, MT, and DNI.