Writing $M_T=\max_{0\le t\le T}S_t\,,\,\, m_T=\min_{0\le t\le T}S_t$ the option payoff is
\begin{align}
(K-S_T)^+\underbrace{1_{\{m_T\le L\}}}_{\text{KI}}\underbrace{1_{\{M_T< U\}}}_{\text{KO}}=(K-S_T)^+(1-1_{\{m_T>L\}})1_{\{M_T<U\}}\,.
\end{align}
In other words, the KI-KO-option is a portfolio of a long positon in a single barrier KO option and a short position in a double barrier KO option.
I made the assumption that there is no complicated interaction between KI and KO. That is:
once the option is knocked in the lower barrier $L$ has done its job and becomes irrelevant;
likewise, once the option is knocked out by $U$, it cannot subsequently by knocked in by $L$ anymore.
Closed formulas for double barrier KO options have been known for a long time. See for example the book by E.G.Haug, Complete Guide to Option Pricing Formulas.