Your actual load is 100W...
With no losses the current is 100W/12V = 8.33A
This means you need 8.33Ah capacity for each hour of running, with provisos as below.
WIth a 7Ah battery that implies 8.33A/7Ah = 0.84 hours
The nominal voltage of a 12V battery is 12.6V, but you will be lucky to have 12V at the inverter terminals.
The battery will be rated at the 10 or even 20 hour rate, so that is a current of 0.7A or 0.35A for a 7Ah battery. The current you are drawing is 10 times that, in reality too much for this battery. The battery capacity is derated considerably, so the implied 100W running time of 50 minutes becomes maybe less than half an hour.
In practice the losses due to the inverter must be added as part of the load. As this is a low load for an 800W inverter it would just be a stab in the dark to say what its losses are, unless you can measure the actual current drawn from the battery when running under that load. My feeling anything from a few watts to 150W losses. It could more than double the load. You could be down to 10 minutes if the battery doesn't get overheated. Batteries are also less capable in cold conditions. A 7Ah battery is more suitable for loads about 10W.
Get the drift? You need a larger battery. The load is more consistent with an 80Ah size deep cycle battery. This would run more like expected, up to 9h, perhaps somewhat less due to inverter losses.
If you want to run the Laptop only, it makes more sense to use an external battery pack that plugs in more directly. This will be more efficient at least. Most laptops run from an 18V battery. Look into the solutions for converting 12V to 18V (Laptop car adaptors), that will run on your battery as well. A solution that might be available, or soon will be, is a fuel cell for laptops. See the link. We are talking about something like 200ml of Methanol for 20h for a notebook. Sounds attractive.