New LOR North extension.

If the LOR was rebuilt as per my earlier blog, then it still wouldn't connect to one of the original destinations of Merseytram, Kirkby.

In it final form the LOR was connected to the main rail network at Seaforth, see this map. Until the 1990s the route was still clear, though now it has been filled in with houses. The LOR's trains usually terminated at Seaforth & Litherland station. During the Grand national meeting, some trains would go to Aintree Racecourse railway station via the North Mersey Branch.

The North Mersey Branch heading east as it crosses the Southport line.

From it junction with the Southport line, there remains a single unelectrified set of rails, maintained by network rail to their minimum standard and used by maintenance vehicles to access the Ormskirk line. A junction connects the remaining line to Aintree station, but beyond the junction one bridge has been removed but the rest of the route is clear to Fazakerley junction on the Kirkby line between Kirkby and Fazakerley.

Extension

In my original plan, I terminated the New LOR at Sandhills station. The track there is now down to 2 set of rails, previously there were many more lines as out to Southport was quad and the North Liverpool Extension Line also passed through, see here. In the intervening stations between Sandhills and the Junction for the North Mersey Branch, the remains of the extra lines can clearly be seen. This would allow extra lines if needed to connect the New LOR from Sandhills to the North Mersey branch. Though as the lines currently only carry four trains per hour, it would be possible to fit additional services in Tram Trains.
Mersey rail has long term plans to reintroduce services on the North Mersey branch to Aintree, with at least 2 intermediate stations. On the map at the top of the page, the line from Sandhills to Aintree is marked in white as it is either in use or Merseyrail has expressed some interest in use/

Beyond Aintree

Merseyrail's plans involve leaving the North Mersey Branch at the Aintree North Curve that might be a reasonable thing, however, it would also be possible to reinstate the line beyond Aintree to Fazakerley Junction.  It would then join Merseyrail's line into Kirkby, taking advantage of the Bridge over the M57, though as this is only a single track it may represent a bottleneck.
At Some point, the line would leave Merseyrail, perhaps after Kirkby station and revert to tram running through Kirkby, where it could be used not only for people intending to use the LOR into Liverpool but also as a feeder for Kirkby station for trains heading East or West. This corresponds to the purple line on the map at the top.
This is an additional 7 km of surface lines over what Merseyrail has either work at the moment or plans to have in use. Which using the costing in the original article come to between £119.21 and £238.41 million, bringing the total for the system to £550 & 1280 million. The biggest disruption in the build would be putting the track down on Kirkby's roads.