I recently brewed a brown ale which due to lack of time and work I haven’t been able to bottle it. It has now sat for over a month, it is well cleared and needs to be bottled. I was just wondering if I will need to add extra yeast to it along with the priming sugar to make sure that it carbonates? And what would be the best choice in yeast? Or should I mix some of the yeast cake at the bottom back in? All of your help it appreciated!
Assuming it’s not a high gravity beer (1.080+) it should be just fine. It doesn’t hurt to add a little extra yeast at bottling though. I usually use a neutral yeast like us-05. Dry yeast is great for this because it’s easier to dose out just half a package or so (and that’s all you need).
Or just pick up a little trub when you rack it and that will likely be enough when combined with the yeast in suspension 
I am a very lazy brewer and I let my beer sit for a month a LOT. Like, most of the time. If/when you rack it to the bottling bucket, just carry over an ounce or two of the yeast cake from the bottom of the fermenter, and everything will turn out great. Now, if you’d have let your beer sit there for two or three months, well then you need to add fresh yeast. But one month? No problem at all, no worries whatsoever. The yeast is still alive down there and ready to eat your priming sugar, guaranteed.
Should be fine.