Cooperation

What makes it easy to work with someone to help them improve themselves?

Using the example of a coach working with a swimmer, it is much easier to help someone...
...when they know they need the coach's help rather than when they think they are doing a great job by themselves
...if they pay attention when the coach is showing them what they need to do to improve
...if they are willing to put the work and focus in to do what the coach prescribes
...when they understand specifically what they are doing wrong and don't like the fact that they are doing it incorrectly, rather than not trying to understand and not caring even if they do understand

Working with a swimmer who is coachable but may not have much inherent skill is desirable to working with someone stubborn and unwilling to improve even if they are very talented.

It seems to me that as long as a person is willing, tractable, and not arrogant, then they can be happily worked with no matter how badly they do.  When you know someone is trying their best, it is hard to give up on them, even if they are floundering a bit.

Despite the fact that Jesus will never give up on a believer, I want so badly to be easy and pleasant for him to work with.  I think repentance is a huge part of coachability because the Bible definition (from dictionary.com, of course!) looks a lot like a simple willingness to do what is right:

"Metanoeo, meaning to change one's mind and purpose as the result of after knowledge.  This verb with the cognate noun 'metanoia', is used of true repentance, a change of mind and purpose and life..."

I think the big trick is to not just to do our best, but to also ask for more help from Jesus because often the things we need to change our mind, purpose and life about are deep habits, maybe even things we enjoy even though they are wrong.  As noble as it would be to do everything out of sheer willpower, I think the whole point of having a Lord is so that we have extra help and support to save us from our dead human nature:

For all have sinned and fall short of the glory of God, 

and are justified by his grace as a gift, 

through the redemption that is in Christ Jesus,

whom God put forward as a propitiation by his blood, 

to be received by faith.

Since we are already justified, we can walk freely with Christ, knowing that we are already good enough on the inside because God said so.  Our right response to that freedom is to desire to reflect our true nature in Christ, and to let him direct us into it.