Return total number of rows and selected (aggregated) data
As I understand the question, you don't need a window function. Aggregate functions do the job:

count() in the lowest level (-&gt; row_ct).
sum() the resulting row_ct in the next level (-&gt; total_row_ct).

SELECT row_to_json(selected_records)::text AS data
FROM  (   
   SELECT array_to_json(array_agg(row_to_json(records))) AS data
        , sum(row_ct) AS total_row_ct
   FROM (
      SELECT landing_path_id
           , sum(entrances) AS entrances
           , count(*) AS row_ct
      FROM   report_la
      WHERE  profile_id = 3777614
      GROUP  BY landing_path_id
      LIMIT  10
      ) records
   ) selected_records;
I also included landing_path_id so the resulting data makes sense.
SQL Fiddle.
Window function?
A window function (count(*) over ()) does not seem to be what you want, since you don't have unaggregated rows.
You could add to the inner subquery:
count(*) OVER ()

.. to get the count of distinct landing_path_id, which is one other possible number that might be of interest. But that doesn't seem to be what you meant by &quot;the total number of rows from that records select&quot;.
Or you could add  to the inner subquery:
sum(count(*)) OVER ()

.. to get the total count with every landing_path_id redundantly, but that would seem pointless. Just mentioning that to demonstrate it's possible to run a window function over the result of an aggregate function in a single pass. Details for that:

How will the order of operations in this query affect my results?
How do I get the aggregate of a window function in Postgres?
Need separate columns that calculate percentage

Updated question
Your result, just without total_count in the records subquery. Now accounting for the LIMIT in the inner SELECT. Even though a maximum of 10 distinct landing_path_id is selected, all qualifying landing_path_id are counted.
To get both in one scan and reuse count and sum separately I introduce a CTE:
WITH cte AS (
  SELECT sum(entrances) AS entrances
       , count(*) over () AS total_count
  FROM   report_la
  WHERE  profile_id = 3777614
  GROUP  BY landing_path_id
  LIMIT  10
  )
SELECT row_to_json(selected_records)::text AS data
FROM  (   
   SELECT (SELECT total_count FROM cte LIMIT 1) AS total_count
        , array_to_json(array_agg(row_to_json(records))) AS data
   FROM  (SELECT entrances FROM cte) records
   ) selected_records;
If you don't care about the attribute name, you can have that cheaper with a subquery:
SELECT row_to_json(selected_records)::text AS data
FROM  (   
   SELECT min(total_count) AS total_count
        , array_to_json(array_agg(row_to_json(ROW(entrances)))) AS data
   FROM (
      SELECT sum(entrances) AS entrances
           , count(*) over () AS total_count  -- shouldn't show up in result
      FROM   report_la
      WHERE  profile_id = 3777614
      GROUP  BY landing_path_id
      LIMIT  1
      ) records
   ) selected_records;
You get the default attribute name f1 instead of entrances, since the ROW expression does not preserve the column name.
If you need a certain attribute name, you could cast the row to a registered type. (Ab-)using a TEMP TABLE to register my row type for the session:
CREATE TEMP TABLE rec1 (entrances bigint);

...
        , array_to_json(array_agg(row_to_json(ROW(entrances)::rec1))) AS data
...
This would be a bit faster than the CTE. Or, more verbose but without cast:
...
        , array_to_json(array_agg(row_to_json(
                   (SELECT x FROM (SELECT records.entrances) x)))) AS data
...
Detailed explanation in this related answer:

Select columns inside json_agg

SQL Fiddle.