gap between connected paths illustrator

When I enter live paint mode and increase the stroke size of my paths, I notice the larger I go with stroke size, the larger a gap there is between paths that are intersecting but which have a large angle between them. Is there any way to make illustrator have these paths smooth out so there's no large gap, or come to a tapered point? Here's an example of the problem (two separate paths I made intersect) and an example of what I would like it to look like (a corner from one continuous path I drew). Thanks in advance!