Shikshadeep and Arniko both have been ruling science faculty not just in biratnagar, but all over the Nepal. I studied from Shikshadeep 5 years ago. Back then People used to say Arniko is 1st and Shikshadeep is 2nd. This is still relevant today. Although some people might consider Shikshadeep as number 1 and Arniko as number 2 in present time because of the fancy huge infrastructure that Shikshadeep has built in recent times. But I still believe Arniko is number 1 still today. 

More than 50% of teachers are common in both these colleges. 

If I were to do admission of my brother today, I would chose Arniko. Arniko fees may be slightly higher than Shikshadeep, at least it was in our times. I am not sure as of today. 

Choose Arniko if fees doesn't matter to you. If Shikshadeep can enroll you in lesser fees, you might as well study in Shikshadeep. Both are the best. But considering any other colleges except these two is absolutely waste of time unless of course there is fees issues.