For the following connection string in my Web.config ;

<add name="WEDOBADGEDatabase" connectionString="Data Source=(LocalDb)\development;Initial Catalog=WEDOBADGE;AttachDBFilename=|DataDirectory|\DevelopmentDatabase.mdf;Integrated Security=SSPI;" providerName="System.Data.SqlClient" />

Where the most important piece of information is Data Source=(LocalDb)\development , [replacing (of course) C:\Program Files\Microsoft SQL Server\120\ with the correct path & use the correct name chosen in your connection string after Data Source=(LocalDb)\ (noncase-sensitive, development in my case)] try:

"C:\Program Files\Microsoft SQL Server\120\Tools\Binn\SqlLocalDB.exe" info Development

This should output something similar to:

Name:               Development
Version:            12.0.2000.8
Shared name:
Owner:              hostname\username
Auto-create:        No
State:              Running
Last start time:    8/2/2016 3:20:57 PM
Instance pipe name: np:\\.\pipe\LOCALDB#CDE5547F\tsql\query

Copy \\.\pipe\LOCALDB#CDE5547F\tsql\query from the "Instance pipe name", WITHOUT the initial np:. Also, your pipe name is likely to be different than mine.

Then you can create a new session in HeidiSQL like this:

N.B.: "Network type", "Hostname / IP", and "Use Windows authentication" are important; whilst other settings (such as "Databases") is up to you, so you could leave them empty / skip, etc.

Credits go to https://stackoverflow.com/a/33748584/11895 for the pipe name suggestion.

As of heidisql.exe Revision 9.3.0.5108 you could also:

1. Start the database with "C:\Program Files\Microsoft SQL Server\120\Tools\Binn\SqlLocalDB.exe" start "MSSQLLocalDB"
2. Read the "Instance pipe name" from "C:\Program Files\Microsoft SQL Server\120\Tools\Binn\SqlLocalDB.exe" info "MSSQLLocalDB" (e.g., \\.\pipe\LOCALDB#1B9DCF1E\tsql\query
3. Start HeidiSQL with "C:\Program Files\HeidiSQL\heidisql.exe --nettype=3 --host="\\.\pipe\LOCALDB#1B9DCF1E\tsql\query" --winauth=1"

I use a Bash script that automates all that, and it goes roughly like this:

shopt -s expand_aliases
alias __sqllocaldb='/c/Program\ Files/Microsoft\ SQL\ Server/120/Tools/Binn/SqlLocalDB.exe'
alias __heidisql='/c/Program\ Files/HeidiSQL/heidisql.exe'
__sqllocaldb start  "Development"
hostname=$(__sqllocaldb info  "Development" | tail -1 | sed 's/\(Instance pipe name: np:\)//')
__heidisql --nettype=3 --host="$hostname" --winauth=1

Here is an AutoHotkey script that automates everything, from copying DB instance name to opening, pasting and connecting to the server.

(Adjust Sleep if it's too fast)

#NoEnv
SetWorkingDir %A_ScriptDir%
#Warn
CoordMode, Mouse, Window
SendMode Input
#SingleInstance Force
SetTitleMatchMode 2
SetTitleMatchMode Fast
DetectHiddenWindows On
DetectHiddenText On
#WinActivateForce
#NoTrayIcon
SetControlDelay 1
SetWinDelay 0
SetKeyDelay -1
SetMouseDelay -1
SetBatchLines -1
#Persistent

HeidiSQLConnectToSSMS:
batch := 
(LTrim
"sqllocaldb start MSSQLLocalDB
for /f ""tokens=3 delims=:"" `%`%i IN ('sqllocaldb info MSSQLLocalDB ^| findstr ""Instance pipe name:""') do set sqlConn=`%`%i
set /p =""`%sqlConn`%""<nul | clip"
)  ; batch
FileDelete, sqlConn.bat  ; sqlConn.bat
FileAppend, %batch%, sqlConn.bat  ; sqlConn.bat
RunWait, sqlConn.bat, , Hide  ; sqlConn.bat
Run, C:\Program Files\HeidiSQL\heidisql.exe  ; HeidiSQL
WinWait, Session manager ahk_class Tconnform ahk_exe heidisql.exe  ; HeidiSQL
Sleep, 333
WinActivate, Session manager ahk_class Tconnform ahk_exe heidisql.exe  ; HeidiSQL
Sleep, 333
ControlClick, TButton4, ahk_class Tconnform ahk_exe heidisql.exe,, Left, 1,  NA  ; New
Sleep, 100
ControlFocus, ComboBox1, ahk_class Tconnform ahk_exe heidisql.exe  ; Network Type
Sleep, 100
ControlSend, ComboBox1, {Home}, ahk_class Tconnform ahk_exe heidisql.exe  ; Select Microsoft SQL Server (named pipe)
Sleep, 100
Loop, 5  ; Select Microsoft SQL Server (named pipe)
{
    ControlSend, ComboBox1, {Down}, ahk_class Tconnform ahk_exe heidisql.exe  ; Select Microsoft SQL Server (named pipe)
    Sleep, 100
}
ControlClick, TCheckBox1, ahk_class Tconnform ahk_exe heidisql.exe,, Left, 1,  NA  ; Use Windows authentication
Sleep, 100
Loop, 2  ; Set Hostname / IP:
{
    Send, {Shift Down}{Tab}{Shift Up}  ; Set Hostname / IP:
    Sleep, 100
}
SendRaw, %CLIPBOARD%  ; Set Hostname / IP:
Sleep, 200
ControlClick, TButton7, ahk_class Tconnform ahk_exe heidisql.exe,, Left, 1,  NA  ; Save
Sleep, 100
Send, {Enter}  ; Connect
Sleep, 100
FileDelete, sqlConn.bat  ; sqlConn.bat
ExitApp  ; ExitApp
Sleep, 100
Return