The landlord, not the agent, is ultimately responsible for maintaining the property.
If he is too busy to do so, then he should authorise his agent to do so without delay.
If he is simply trying to avoid spending the money, then he shouldn't be in the property business (for that is what it is).
He is a doctor, he does a demanding job and is well rewarded for it. Why does he think that you would happily pay for a service you are not receiving?
Being an absentee landlord is no excuse; he still has moral and legal obligations and his appointed agent should be working to his instructions.
As per my previous post, it is my view that you should get out of this house and away from this landlord/agent combination. At the end of the day, it is you who decides the direction of your life, however, and only you can decide if your pet is worth the grief when renting, and whether you should be prepared to pay more as a consequence.
Many landlords that say "no pets" take that approach because pets frequently end up costing large sums of money in increased wear and tear despite reassurances from the tenants. (You sound like a dream tenant to me because so many are simply not prepared to take reasonable care of the landlord's investment). However, many landlords will soften their stance in exchange for an increased deposit and/or a modest increase in rent. This won't solve your present problems but could help to make other places available to you.
Worth a try!