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I am certainly no expert on the subject but I would suggest the following points as answers to your question.

1) Most (that I have heard of) plans for Land Value Tax do still include a smaller portion of property taxes. This does not completely solve the problem you pose but does ameliorate it a little.

2) Some of those infrastructure things are paid for via utility payments on utilities used. The 100 families in the apartment building will use much more water than a single family home. Some part of that money should be budgeted and used for improving the utilities available. That doesn't account for things like parks, fire department, etc. but does take up some of the slack.

3) lastly and most importantly those homes will be paying more than their fair share on purpose! If they live in a neighborhood where it is economical to build 100 unit apartment buildings then a lot with just a single-family home is being wildly underutilized. The higher tax costs will encourage owners to redevelop those properties into apartments or other higher value uses to make efficient use of that land. This is the entire point of Land Value Tax! It makes it in the interest of land owners to upgrade and improve their property (whereas traditional property taxes have a negative incentive because building a new building will increase your taxes).

edit: formatting

There are plenty of ways of handling this; in New Zealand, local council rates (like a property tax) are used to pay for infrastructure, and after the city budget is set, each property owner pays in proportion to the value of their property (currently land + capital improvement). This can be separate from the ideally national land value tax, and take into account different infrastructure requirements.

The other reality is that the single family homes in the area which just got a massive boost in infrastructure spending are benefiting from those improvements disproportionately; their land could be providing access for 100 families to those services but instead are only allowing access for 1 family. Their land value will have skyrocketed hugely because of the improvements around them, so they should pay the tax to return to society the land value they gained.

If it doesn't make fiscal sense for them to pay that tax, then they can sell their now hugely expensive piece of land for a tidy profit and live somewhere it makes more sense for them to be, allowing their land to be developed into a 100-family apartment.

> What is the justification for different families having to pay such different amounts for the same infrastructure just because some live in denser housing?

Because this premise is wrong:

> If the tax is just based on the land value won't each of the 100 lots pay the same amount? To pay for the need for 10x as much infrastructure the tax rate will have to up by a factor of 10. The result is that families in the single family homes will be paying per family about 100x as much for the same infrastructure services as those who live in the apartments.

Energy = force * distance = mass * acceleration * distance. People in single family homes on their own quarter acre lot will use a lot more infrastructure than people in apartments, simply because less dense living requires mass to travel much farther.

Namely, single family lots make public transit not viable (simply because individual cars are more convenient and so people will prefer and politically support car infrastructure). And the knock on effects leader to ever more consumption of infrastructure, such as more roads, more parking lots, bigger lots (since driving makes traveling those distances easier), which then leads back to more infrastructure to push more mass more distance.