I think there is a problem of scope confusion here, where the designer might be thinking that a lot of work has been completed for the money paid and in client's perception, the work done is not enough. As you said that the designer is a well known designer so I am assuming that they would have a professional setup.
You can probably take the approach below.
1) Visit the office of the designer and meet a senior executive to initiate a resolution.
2) For the area that you have, write down all the work that has been completed or is still pending. Once all the items are documented, you would know the amount required to complete the work.
3) If, for a particular work, you and the designer have not agreed to a price before, check the price with 2 to 3 of the competitors companies and use an average price for arriving at this number.
This is usually the bone of contention as the prices and the quantity of work is not defined for some items and this causes undue confusion at a later stage.
This would give you a starting point for the negotiation. So,
A) If you do not wish to pay more money then they should complete the work for the amount that has already been paid.
B) You may also choose to pay him a little more money if you wish that he completes the additional items as identified by you during discussions.
Once the quantity of work is defined and the price to execute the work is defined then you should be able to resolve this issue.
Repost, if this does not work.