How long does it take for Elo rating to reflect changes in player's strength? Suppose a player with ELO rating X has improved (for simplicity, without playing in tournaments) and she is now actually playing at the level X+N. How many games will it typically take for her rating to reach the level X+N, depending on N? Is there a quantitative research into that?
The way FIDE rating (and rating change) is calculated is described in FIDE Rating Regulations effective from 1 July 2017. For an already rated player the calculations for working out the rating change are given in section 8.5.

From here it can be seen that how fast a rating changes for a given performance depende on the "k factor". This is defined in 8.56:


  8.56 K is the development coefficient.
  K = 40 for a player new to the rating list until he has completed events with at least 30 games.
  K = 20 as long as a player's rating remains under 2400.
  K = 10 once a player's published rating has reached 2400 and remains at that level subsequently, even if the rating drops below 2400.
  K = 40 for all players until their 18th birthday, as long as their rating remains under 2300.
  If the number of games (n) for a player on any list for a rating period multiplied by K (as defined above) exceeds 700, then K shall be the largest whole number such that K x n does not exceed 700.


Here are the relevant sections of 8.5 describing the calculations:


  8.5 Determining the rating change for a rated player
  
  8.51 For each game played against a rated player, determine the difference in rating between the player and his opponent, D.
  
  ...
  
  8.54 A difference in rating of more than 400 points shall be counted for rating purposes as though it were a difference of 400 points.
  
  8.55 (a) Use table 8.1(b) to determine the player’s score probability PD
  (b) ΔR = score – PD. For each game, the score is 1, 0.5 or 0.
  (c) ΣΔR x K = the Rating Change for a given tournament, or Rating period.


As to your question:


  How many games will it typically take for her rating to reach the
  level X+N, depending on N?


Table 8.1(b) is non-linear so as well as depending on the k factor the number of games also depends on N in a non linear fashion. So, you will have to perform separate calculations for different values of N.

Here is an example for N=100. How many games would a player require to draw against players on average 100 points higher to raise the rating by 100 points?

In that case, according to table 8.1(b), PD = 0.36. So, ΔR = 0.14 and the number of games required for a 100 point rating gain = 100 / 0.14 / k.

For k = 10 number of games = 72
For k = 20 number of games = 36
For k = 40 number of games = 18