First note that $l,m,n\ne0$. This is because, say, if $l=0$, we will have $m+n=0,mn=0\therefore l^2+m^2+n^2=0\ne1$.
Now, the angle between the vectors whose direction cosines $(l_1,m_1,n_1),(l_2,m_2,n_2)$ are given by the equations is equal to the angle between the vectors $(1,m_1/l_1,n_1/l_1)\equiv(1,x_1,y_1),(1,m_2/l_2,n_2/l_2)\equiv(1,x_2,y_2)$, provided $l_1l_2>0$.
$$\displaystyle\cos\theta=\frac{l_1l_2+m_1m_2+n_1n_2}{\sqrt{l_1^2+m_1^2+n_1^2}\sqrt{l_1^2+m_1^2+n_1^2}}=\frac{l_1l_2}{|l_1l_2|}\cdot\frac{1+x_1x_2+y_1y_2}{\sqrt{1+x_1^2+y_1^2}\sqrt{1+x_2^2+y_2^2}}$$
Divide $l+m+n=0$ by $l$,$$1+x+y=0$$
Instead of juggling $3$ constants $p,q,r$, take $q-r=A,r-p=B$ and divide $\displaystyle\frac{mn}A+\frac{nl}B-\frac{lm}{A+B}=0$ by $l^2$,$$\displaystyle\frac{xy}A+\frac yB-\frac x{A+B}=0$$
Eliminate $y$ to get,
$$\displaystyle\frac{x^2}A+x\Big(\frac1A+\frac1B+\frac1{A+B}\Big)+\frac1B=0$$
Sum of roots,$$\displaystyle x_1+x_2=-A\Big[\frac1B+\frac1{A+B}\Big]-1$$
Product of roots,$$\displaystyle x_1x_2=\frac AB$$
$\displaystyle1+x_1x_2+y_1y_2=1+x_1x_2+(1+x_1)(1+x_2)=2+x_1+x_2+2x_1x_2=\frac{A^2+AB+B^2}{B(A+B)}$
$(1+x_1^2+y_1^2)(1+x_2^2+y_2^2)\\=(1+x_1^2+(1+x_1)^2)(1+x_2^2+(1+x_2)^2)\\=4(x_1^2+x_1+1)(x_2^2+x_2+1)$
Substitute for $x_1^2,x_2^2$ from the quadratic equation,
$=4\Big[1-\frac AB-Ax_1\Big(\frac1B+\frac1{A+B}\Big)\Big]\Big[1-\frac AB-Ax_2\Big(\frac1B+\frac1{A+B}\Big)\Big]\\=4\Big[\Big(1-\frac AB\Big)^2-A\Big(1-\frac AB\Big)\Big(\frac1B+\frac1{A+B}\Big)[x_1+x_2]+A^2\Big(\frac1B+\frac1{A+B}\Big)^2x_1x_2\Big]\\=\frac4{B^2(A+B)^2}\Big[(B-A)^2(B+A)^2+\frac AB(B-A)(A+2B)(A^2+3AB+B^2)+\frac{A^3}B(A+2B)^2\Big]$
We have $3$ terms in the sums. I simplified the last $2$ terms first because they have more in common.
$\displaystyle=\frac{4[A^2+B^2+AB]^2}{B^2(A+B)^2}$
We get $\displaystyle\cos\theta=\frac{1+x_1x_2+y_1y_2}{\sqrt{1+x_1^2+y_1^2}\sqrt{1+x_2^2+y_2^2}}=\pm1/2\therefore\theta=\pi/3,2\pi/3$.
In any case, since we are talking about lines which extend in both directions indefinitely and not vectors, the angle between them is often stated as the acute angle between them, given by $\pi/3$.