She sounds like the kind of person who quite enjoys the fact that you are put out by their arrangement tbh.
I'm quite concerned for you in this situation OP.
How long has your DP been out of work?
When do you expect him to go back to work?
Does he have a job to go back to?
How will his return to work impact your own job and his time with the kids?
In my mind you should be establishing the point at which he is able to return to work and how that will affect everything. Is this a short term or a long term problem?
You need to make clear to your DP that his dependency on you to support his children is not a long term solution. You also need affirmation all round that you DO matter.
If this is a short term problem and he will be returning to work soon, I would still address the driving point but do so in the wider context of parenting (eg taking the kids on trips, to friends houses etc). Don't link it directly back to you or else it will just get the exes back up. Give other examples where it has been a hindrance to back up the demand as well (I'm not undermining the importance of you going to work at all).
If this is a long term problem OP, it's a huge commitment for you to be making for what seems like little support or appreciation in return. If I were in your position I would be looking closely at what I wanted from the relationship.
Very best of luck to you, i hope you manage to find the best outcome for yourself and the kids.