Am I responsible for a car that broke down while storing it for a relative?

If your daughter did not have permission (or  instructions) to drive the car (e.g. it was just supposed to be parked, not driven), then you are likely responsible for the damage if she is a minor (if she's an adult, she may be personally responsible if she took it without your  permission). When you use someone's property without permission, you are generally responsible for the consequences--even in the event that, as you say, the engine was likely to seize at some point, since the critical fact is that it seized now, during unlawful use.

If  your daugher had permission or authority to drive the vehicle, then you or your daughter would only be resonsible if she had done something wrong to cause the damage.

The above is in regards to your brother. In regards to the new landlord, if you do not have the right to put a broken car on the property, he can make you move it--you'd have to pay for the move, though you could then, if you were not responsible for the damage, seek to collect the cost back from your brother. However, the important point here is that regardless of how responsibility is apportioned between you, your daughter, and your brother, if you there is no right to park a broken car on the property (e.g. you don't have a parking spot or garage you can put it into), then you'll have to move it. The landlord does not have to allow it on his property unless you have paid for a space to put it into.