Second there are many reasons why smoke doesn't go up the chimney, it might not be just a case of cleaning. One of the main reasons I've discovered is the wind blowing down the chimney. The spark arrestor that sits above the chimney is often just the "T" type and with winds coming from different directions it can completly overpower the draft of hot air rising.
The best caps are the "Spanish Helmet" style that have only one opening and a vane to keep the opening away from the direction of the wind. The worst are the turbine ventilator covers. These are not intended for chimney application. They are meant to vent warehouses where it is warm air not hot air. The bearings get so noisey that your neighbors will hate you late at night from the screeching!
Find out if the fireplace by not being cleaned, would cause potential hazard, If so, present this information and he will likely want to absorb the cost.
Check your lease and see if there are any clauses regarding maintenance of the unit.
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